Novel mRNA technology and (by definition) no long term safety data yet. Any pilot who can Notice, Understand & Think Ahead might want to conduct a very careful risk-benefit analysis before rushing into these vaccines, given the fragility of a Class 1 medical.
mRNA vaccines do not play with your DNA or alter it in anyway. The Pfizer vaccine creates a replicating 'foreign' protein molecule that your immune system sees as foreign and activates. Your immune system then does everything naturally, no vaccine assistance at all, all it's doing is being say the rabbit at a greyhound race, getting all your bodys defenses to race in one direction and seek out that protein. How it works vs Covid, the same protein is found on the surface of Covid virus, so it teaches your immune system to latch onto that protein and see it as harmful and foreign.
Astrazeneca is a vector vaccine, so its basically antigen cells from a chimpanzee grafted onto an inert virus that stimulates your immune system to react, it focuses on a similar protein as the lure/activator. A difference in the AZ method is that the inert virus actually stimulates your antigens into a little overdrive as well as providing a cue for resistance.
Both vaccines work differently, but achieve the same thing, activating your immune system to do its's own thing. The vaccine in itself is not defeating Covid, your own immune system is.
Medical wise, FAA yes grounds you for 48 hours following any Covid vaccination, after then side effects determine return to work. I know a number of US pilots who are now grounded due to covid directly, particularly due to lung issues. Also the well known case of the BA pilot who died from it after a 4 month battle.