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.