Why can't database session driver use a persistent connection?

Got any sources for that MySQL recommendation?
The PHP link has got comments dating back 15 years - and I'm trying to understand if this is a legacy problem that's now resolved.
It seems that it's simple to mitigate with the register_shutdown_function().
On an Azure + SqlServer site I've seen response time drop from 1s to 45ms by using a persistent connection but CI forces me to use file for sessions.

It's not a legacy problem, it's a technical limitation.

Locks are meant to prevent concurrent reads/writes (depending on the lock type) to the same resource, and the only way to identify concurrent requests is by the connection ID that they get. A persistent connection breaks that because it is shared between processes and thus they all share the same connection ID.

But even the reason why you want to use a persistent connection is a trap ... You're getting that reduced response time only because you're the only user at that time. Concurrent requests won't be able to asynchronically share that persistent connection, because the connection itself would be (ironically) locked to a single process at a time, and therefore each request will have to wait its turn to use it - whether it's 5 or 5000, they'll all be processed sequentially, so you'll have a huge bottleneck on your hands, not a performance improvement.
There may be some overhead when each request/process opens its own database connection (the reduced response time with a persistent connection is because it doesn't need to authenticate and do a ton of other initialization procedures each time), but that allows them to work simultaneously in isolation of each other.

As with everything else in programming, it's a simple trade-off - you pay a small price upfront, for significant gains later.

The difference between 1s and 45ms is unusually large though, it should certainly be reducible by other means. First thing I would try is putting the database server IP address (instead of hostname) in the configuration - that skips a DNS lookup that might be slow.