The navy do get training in small boats, we see them practicing whenever we leave Portsmouth on a day time ferry bound for France.
But the thing about the RNLI is that their crews are local, many have links to the sea, as fishermen and the like. It means all the crews know their local waters and coastline very well indeed, and when you may need to get close to shore, when rocky outcrops reduce water depth to inches and could tear a life boat's hull out, then you need crews who know their territory.
We are RNLI supporters and get their magazine and so many of their inshore and offshore rescues, some in horrendous weather conditions, are successful because the crews know their territory so well.