Yes, though a few providers offer notification, it’s far from universal and AFAIK not legally required at the provider level. Hell, Flowroute doesn’t mention notification in their current documentation, even though they are owned by Intrado!
A business is unlikely to switch trunking providers just for E911 notification. Using a different provider just for 911 is also problematic, because you aren’t regularly sending them traffic so you know the trunks are working.
Some providers offering 911 require that the DID be with them, i.e. you either must use them for origination or have the admin hassle of two DIDs per user.
If you have failover, e.g. in case of an internet outage you have one POTS line for 911. It may not send the proper address, but it’s a lot better than nothing. Especially in this case, notifying local security is important.
This is not a simple problem and doing notification in the PBX is IMO a cleaner solution.