In theory, Search for Azcanta seems more dangerous in a larger card pool. You have a wider array of high-value cards to find, plus having an extra mana source lying around is potentially more impactful. Additionally, Search for Azcanta is much easier to transform in Modern thanks to Thought Scour and the fetchlands. It's almost trivial. Think about how easy it is to cast Gurmag Angler on Turn 2. If you are hell-bent on transforming Search for Azcanta the turn after you cast it, you should be able to make that a reality.
You would ideally be finding cards to help you lock up the game or end it. Discard is rather poor because your opponent will likely be empty-handed, so you're probably looking for counterspells. Remand and Mana Leak get worse as the game goes on, but Logic Knot, Cryptic Command, and Spell Snare don't.
Finally, you should be able to find a win condition.
My hope is that Search for Azcanta revitalizes combo/control, which is one of my favorite archetypes to play and tune.
Over the course of Modern's history, I've registered each of the above win conditions. There is just something appealing about a win condition that isn't strictly a win condition. Aside from Scapeshift, each of those cards can double as a defensive tool. Additionally, they all kill relatively quickly, which is a requisite for Modern control decks.
The first thing I'll note is that Search for Azcanta isn't necessarily what these decks needed. However, it does slot in nicely in each deck, and does things each deck wants to be doing. Realistically, what these decks need is to be a tad bit faster, and while Search for Azcanta is a Rampant Growth, that sort of mana acceleration isn't exactly tearing up Modern.
Chrome Mox was what made old Thopter Foundry decks truly viable, as it allowed them to keep up with the faster decks in the format. The problem doesn't lie in the fact that they don't have adequate mana acceleration; the other decks in the format are simply too fast and that makes your other defensive cards, such as Remand, not capable of doing the job they are meant to do.
I turned to Timely Reinforcements as a sort of maindeck hate card for any decks that were trying to kill me with creatures, and that's worked surprisingly well in Jeskai Control with Nahiri, the Harbinger and the Thopter Foundry decks I worked on. Timely Reinforcements also plays quite well with Search for Azcanta and the combo-esque win conditions, as with both, all you need is time before the game is effectively over.
Search for Azcanta doesn't automatically make these decks viable, but it is a valuable addition that makes them much better.
Search for Azcanta does little to alter traditional Scapeshift. It takes the place of the third ramp spell and adds some staying power, but that might be enough to increase consistency and power the deck up a bit. You would think having a one-card combo would be all upside, but it just means your deck doesn't do anything when you don't find it. Given enough time, Azcanta will get you there.
This deck could play Thought Scour, but that can be dangerous. When you need to Scapeshift for Mountains and Valakut, the Molten Pinnacle, you actually need to have those cards in your deck. If you happen to mill over your two copies of Valakut, you're going to lose. Hitting both Valakuts isn't likely, but I am concerned about not having enough Mountains to kill my opponent once I hit seven mana. Additionally, Thought Scour can help transform Search for Azcanta sooner, but it's not a necessity.
Explore gets the nod over cards like Farseek and Search for Tomorrow because this is a velocity-based deck centered on a combo kill. Drawing a card and playing an extra land from hand is much better than ramping and still having that extra land in your hand. You need to play slightly more lands in order to make that situation come up frequently with Explore, but it's something I'm fine with. Scapeshift wants to develop its mana first and foremost and dig for action later.
The Izzet Charm, Electrolyze, Kozilek's Return, and Snapcaster Mage could basically be anything, as long as they are defensive cards. Red doesn't have the best options, as you're fairly limited to smaller burn spells, which means sometimes you'll get run over by a Reality Smasher. The hope is that cards like Remand, Cryptic Command, and outright killing them with Scapeshift will be good enough, and it's mostly true.
I wouldn't want to load up on removal that can't double as velocity, which is why Izzet Charm and Electrolyze make the cut. Kozilek's Return could be an Anger of the Gods or Firespout, but there aren't many three-toughness threats at the moment, so I prefer the instant speed.
In this version, Prismatic Omen is a control tool and eventual win condition alongside Valakut, the Molten Pinnacle. You could play more Scapeshifts to enable faster kills, but winning with Prismatic Omen happens relatively quickly. Instead of focusing on comboing people, I wanted to see if I could build a more controlling version.
Wargate is excellent and could potentially have a more prominent role in the decklist, but I worry that it's too slow. In order for it to truly shine, you need to be playing matchups where your biggest concern is consistency and threat density, say, against something with discard and Abrupt Decay, but those aren't my biggest worries in Modern at the moment. At the very least, you can Wargate for sideboard hate like Stony Silence and Rest in Peace, so it's still very powerful against some of the faster decks.
Thought Scour has a place in this list because you don't care about milling anything specific, unlike the dedicated Scapeshift lists. You need a Prismatic Omen to win, but you shouldn't be concerned about milling all of your copies when you're playing four. That said, I'm still not playing Thought Scour because you don't necessarily need to transform Search for Azcanta as soon as possible.
You could potentially merge the two lists and include Prismatic Omen and Scapeshift, but I don't feel like that's necessary. Playing white gives you a host of excellent options in Modern, whereas if you wanted to add the requisite Mountains to make Scapeshift lethal, you'd be taking a ton of damage from your manabase.
Straight Temur Prismatic Omen / Scapeshift could potentially be great, but I wanted to try a Bant list with Wargate Prismatic Omen as a control tool. Temur would lose out on the white options, but the red replacements might be good enough.
This is a personal favorite of mine and one that I'll probably continue to work on in the background. Hopefully others give it a try also.
One of the biggest questions to ask: why play a controlling version of Thopter Foundry when you can use the one with Whir of Invention and Ensnaring Bridge? Unfortunately, I don't have a good answer for that. Oddly enough, that version of Thopter Foundry is probably a worse version of Lantern Control and I can think of a few reasons to play this deck over Lantern, so maybe it has merit.
Realistically, you can play Search for Azcanta, Thought Scour, and Logic Knot in any Modern control shell and it will be reasonable. How you choose to end the game is up to you. Of the options, Thopter Foundry is probably not the best one, but it is a viable choice, especially if most of what you're facing is creature strategies.
The one glaring weakness of Thopter Foundry is that it doesn't kill very quickly, which further exacerbates its weakness to combo decks. This deck should probably only be played in a creature-heavy metagame. Cards like Time Sieve and Krark-Clan Ironworks can combo with Thopter Foundry / Sword of the Meek to defeat your opponent, but they require even more setup. Time Sieve is a reasonable option, especially if you play more Muddle the Mixture, but you're often better off just having more disruption.
Any blue deck without black is going to struggle against the combo decks that can power through counterspells, as you typically need the one-two punch of discard plus counterspells to defend yourself against them. Discard isn't the only avenue of attack, but you do need something proactive like a hateful permanent, as sitting back on counterspells (especially soft counterspells) won't win you the game without a clock.
I played Jeskai Control with Nahiri, the Harbinger to good results in a couple of tournaments, but then it seemed to drop off the face of the earth. Search for Azcanta is being looked at as a replacement for Nahiri, the Harbinger in Jeskai Control, but I haven't seen anyone think about putting the two together.
Again, the necessity for hard counters shows up here, especially since Jeskai is more of a control deck. Your plan is to buy some time with things like Timely Reinforcements and win the game with Azcanta or Nahiri. Having access to cards like Logic Knot make it more likely that you actually keep control of the game once you have Azcanta going.
Pete Ingram's take on Jeskai with maindeck Spreading Seas is completely reasonable. It's a fine thing to be doing early if you don't think anything else is going to be happening. He was disappointed with Search for Azcanta in Jeskai, but he didn't try different incarnations of the archetype.
Bonus Esper Control Decklist
Cards like Batterskull and Lingering Souls are sweet cards that play defense, double as win conditions, and triple as things you can find off Search for Azcanta. They are incredibly slow at actually winning the game, but thankfully this deck should actually be able to take full control.
Using Search for Azcanta to find your Esper Charms in the mid-game basically amounts to a soft lock. You won't necessarily lock up the game with a fast win condition, but you can certainly make sure they aren't getting the most out of their draw steps.
It's only a matter of time before Search for Azcanta makes its presence known in Modern. With a metagame full of Storm, Affinity, and Tron, Search for Azcanta has its work cut out for it, as the archetypes it slides into has a difficult time with those sorts of decks. Should the format slow down again or shift into a world with Thoughtseize, Search for Azcanta probably will be great.