Not really. mRNA starts to degrade within minutes of injection, and lasts maybe at most a few days in your body. It cant turn back into DNA, so once its gone, it can no longer have any effect on you.
> Why has this not been approved before?
Its a relatively new technology that took a lot of work and money to make a stable, effective vaccine with. Its also relatively expensive to manufacture.
> Would an mRNA vaccine be approved under less pressing circumstances...?
Yes, though these almost certainly were approved faster than other vaccines would be because of the pressing nature of things.