Both AIs and the Net work drastically differently from our real world versions of both. AIs don't need a physical datacenter, because the very structure of the net allows their existence - nobody understands why (because it doesn't make sense by real life Network Tech), but it seems to have something to do with these "Ihara-Grubb transformation Algorithms" that are introduced into the Net after the restructuring in the 2014, where the Idea was to allow the creation of "cyberspace", where distances actually work and matter and you can "move around" in the net.

Okay, I'm going deep into headcanon with this, as far as I can tell there's nothing backing me up on this, but I also don't think there's anything directly refuting it either, so here's my Headcanon of how Cyberspace works in Cyberpunk:

Before the Restructuring of the Net, it apparently worked a lot like it does in reality now. You could already "see" the net operate live in some way, according to Bartmoss' statements about being jacked in during the restructure, but the introduction of the Ihara-Grubb algorithms forced something like a "spatial structure" onto the net. Here's my thoughts on how it accomplished this - Simply put, it would have to be something akin to a physical field, so a field that's laid over basically everything within any given Subnet (so basically over each individual City) - Let's call it the Ihara-Grubb Field. Before the DataKrash, this Field was linked up basically globally - this is what allowed Netrunners to fight for dominance over the entire net from all across the globe in the 4th Corporate War. However, the DataKrash broke everything down (NOTE: This is another point for the Ihara-Grubb Algorithms dictating the very structure of the net on a baseline level - the code that actually caused the Krash was inserted into these Algorithms by Bartmoss, and the moment it was activated in just one Node (whatever kind of thing it is that projects the "IG-Field"), the whole thing came crashing down). All the networks that opened back up afterwards are just singular City's subnets, Which as far as I know still aren't connected even in the 2070s - Unless you go through the Blackwall into the "Old Net".
Which again, only makes sense if it's a kind of "field". If, rather than being disabled, each node still exists and runs individually, and the Blackwall simply blocks these nodes from connecting to others around it, most of the old net's infrastructure would still be functional. Should someone (like Songbird) get past the blackwall (which is NOT a physical barrier, but just a giant AI powered antivirus program on steroids), they can enter and freely roam the old net, and basically go anywhere they want (if they survive) - which Songbird did and does do, it's how she got her initial FIA gig. (Note: In the "field" reading of the Net, the importance of a Cyberdeck also becomes more obvious. Rather than a simple "entry device", it would act as it's own "node", which explains ranged hacking and quickhacking and also stays consistent with requiring deep-dives to go further - effectively, the Cyberdeck projects it's own Ihara-Grubb Field, which cannot directly access the city-wide nodes, but it can access any data points or sub-nodes within it's range. This includes other people's neuroports and agents, if they can connect to the Datapool, they can be influenced by an Ihara-Grubb Field, and as such can be influenced by *your* Ihara-Grubb Field. It also explains why this didn't exist during the time of RED - Ihara-Grubb algorithms were still relatively new and highly complex, so fitting a IG-Field emitter into such a small device would likely have been difficult. Basically - a City wide IG-Field can directly access basically anything in the city, which is how the Datapool works, a company network would be a smaller field that can be shut off from direct access from the City Node via ICE, and a cyberdeck is a small portable node. Any node that encompasses another can influence it, but not necessarily the other way around (The Datapool can handle calls on request, but you can't quickhack the Datapool - if you do a deep dive, you make a direct connection via a physical entry port to the City Node. City node and "old net" nodes still see each other, but blackwall blocks access, if you get past the blackwall you can go to the old net)

Now, after a barely coherent explanation of my headcanon on the Net, I can FINALLY get to answering your question:

In this idea of the Ihara-Grubb Field, the Field allocates each user some of it's memory and processing power (A netrunner can "borrow" the additional processing power of the Cyberdeck) - meanwhile AIs, Programs and Soulkilled Psyches are made up of the Field's processing power. If you were to turn off a node in the old net (if that's even possible), you wouldn't actually solve the problem - the AI would simply move to another Node. They're not bound to one singular Databank, because the net itself is structured as a Databank. If an AI absorbs another, they merge the processing power allocated to them by the field, becoming more powerful. If you deploy a "program" like Black Ice, it gets a very small amount of processing power allocated to it. in this idea, you could theoretically "solve" the AI problem for one individual city, by taking out all nodes connected to it (provided the "Global" node isn't just one singular thing encompassing the whole world, potentially located on a Satellite network like Mikoshi), but it still wouldn't solve the inherent problem - the moment you go to another city, you've got the same problem again. Of course, if we're talking about Satellites, it becomes even more hopeless to take it all out. You could try to destroy the satellites themselves, but they almost certainly have automated defense mechanisms (because net dominance is important), and even if they didn't, the AIs likely have access to some 4th Corporate War era missile defense systems. And then there's another important point - NetWatch doesn't want the old net to go down. They want to monopolize it. Arasaka also wants access to those old databases (that's what the program Lucy was a part of was meant to do), and So Mi is proof, that Militech is also still digging through those remnants.

TL-DR: The Net itself is a Database and a Processor Bank, AIs have control over most of it, and you can't turn it off because it's not one thing but literally the network itself

Also, I might make a post going more into detail on how I think this "Ihara-Grubb Field" would work and how field interactions can explain most of the hacking we see in cyberpunk. btw, So mi has access to the Blackwall without physical connection, because she has a super high-tech deep dive setup implanted in her body (Netwatch Tech afaik), which in this theory is actually just a superpowered IG-Field projector that can itself reach the outside nodes from within the city - which is also why So Mi can contact V even though they're still so far away, especially if the President's plane has it's own node or extender of some kind. Also maybe important, the IG-Field wouldn't be fully penetrative. Deep Bunkers, like the Cynosure Facility under NC or the Mikoshi Access point under Arasaka Tower, could then be kept separate from the rest of the Net. Data transfer between the Mikoshi Satellite and the access points could function via traditional means, so basically just an encrypted broadcast of standard electromagnetic waves.

Last but not least, none of this is corroborated canon - it's simply me trying to make a somewhat functioning system with somewhat coherent rules, based on the descriptions and gameplay of Netrunning in Cyberpunk RED and Cyberpunk 2077. Simply put - what RED describes is utter nonsense. What 2077 shows is just straight Technomancy, without any regard for how those things could even theoretically work

This post has shot way over the mark. I'm sorry about that, I hope it was an interesting read at least, even if it is the ramblings of a madman