This is incorrect. When I worked on LTE networks, I did work on a few problems with VoLTE based 911 service, although I didn't specialize in 911 services.
In many ways, something like an LTE tower doesn't really know what a phone call is. It just looks like packet data. So there are several fields in the signalling, that indicate that a request is related to an emergency, which should have emergency handling. This can include things like dropping other lower priority traffic, prioritizing the packets themselves on the network, etc.
But the other interesting thing is in VoLTE, you actually do a new IP assignment (to an sos PDN) specifically for the emergency call, so you have a sort of new registration to the network. This allows you to make an emergency call on any network that is available. Although scanning for an available network can take some time, so you'll likely use your own carrier if they're currently available.
Note: I've been out of the industry a few years now, and specific to emergency calls only have knowledge specific to LTE and UMTS.