Precisely the reverse: attenuated virus (live virus) vaccines regularly allow mutation back to pathological forms. See: polio. This is because an attenuated virus is still replicating viral particles and every replication is a lottery ticket that the virus might mutate back.

mRNA vaccines, by contrast, causes no viral replication so the virus doesn't get these lottery tickets.

Of course, either type of vaccine is going to place evolutionary pressure on a disease to evolve an escape. That's just what happens.

One side note: one of the things about mRNA vaccines is that they can be very specific. Now, that may not be great for Covid where we would like to be more general. However, that may be absolutely essential against Epstein-Barr where a chunk of the virus looks like nervous system proteins. A standard vaccine is likely to prime the immune system just like the virus and cause multiple sclerosis as a side effect. An mRNA vaccine, by contrast, could target only those sections of the E-B virus that look sufficiently different from the nervous system and prime the immune system without risking friendly fire.

Why would a live vaccine be better?

I feel like the parent wasn't trying to argue that injecting people with live vaccines (er, live virus?) would be better?

Rather the idea seemed to be to stop making more vaccines for COVID - kind of like how we deal with colds now.

There would be cold vaccines if it were possible to produce effective ones, so it's not an argument against producing a highly effective vaccine.

They were explicitly saying their concern was with non-live vaccines though?

I obviously can't say for sure, but given the topic was the development of artificial resistance to vaccines, I would've thought the alternative they had in mind was natural resistance - i.e. getting infected naturally.

Just in my non-expert head - a live vaccine is using the virus as it already exists to confer immunity - not trying to code against a ton of hypothetical variants which could select for more radical variants

Ah, that isn't how a live vaccine works though. They get attenuated (aka damaged/killed/broken) or similar but not the same viruses (aka cowpox used to be a live vaccine for smallpox), then introduced. The body reacts to the not-really-functional virus, but it isn't the same as if it was an actual infection.

It still triggers the immune system, but mutations in any of the major receptors or the like will cause it to be less effective too.