Given oath's laughable privacy policy, I'm not sure any stepping would be required.
Granted the shutdown came before the Verizon/Oath takeover, but somehow I doubt the decision came down to "we could make a lot of money off this, but it will require relaxation of privacy practices, so we'll just shut it down instead."
If you read blog posts and the like from people involved in those projects (MSN Messenger and AIM/ICQ, specifically); the companies would take advantage of protocol updates or other scheduled downtime to failover to newer hardware. So I doubt they were ever stuck on any “obsolete hardware”.
The original AOL client shipped for DOS with Digital Research's GEM widget kit, it wouldn't surprise me if they used additional DR hardware and software on the backend (VAX anyone?). When I worked on the AOL project, I would see e-mails from AOL ops (from AOL screen names, no less!) about how they would take scary and unique sounding things like "access rotors" down, so I suspect that given how early the launch was, the backend was extremely bespoke simply because they were making things up as they went along in order to launch a new and unique thing.
It wouldn't surprise me one bit to learn that it ran on VAX, HP3000, AS/400 or perhaps something even older and larger. You didn't hear about horizontal scaling nearly as much back then, most scaling consisted of "buy a bigger computer."
I think the thing to keep in mind is that running centralized backend infrastructure (ie; store and forward, user state, directory) for an instant messaging service is not cheap and if the platform is open and allows for user supplied clients, there's almost zero direct monetization opportunity without stepping into unethical privacy breach territory. That means they're relegated to just pushing a brand, and it seems like a very expensive way to do that. Moreover, AOL transitioned in its later years into a digital journalism company, and as far as I know AOL itself as a consumer brand disappeared.
[0] Worked on a whitebox set top box browser platform (ala Android) in the early 00s specifically with AOL as a client. Amusingly, the big feature they were trying to push was onscreen chatrooms for people who were watching the same thing live, which I guess if you think about it may have led to ways to monetize chat.