I would not do a bay window and bench, that is too dated IMO and not worth the money. Remove the corner shelves. Just get yourself a round table and chairs and be done with it.

You have already made up your mind in the last two sentences, negating what you first said. Go with it! One piece of advice would be to install double hung windows which will eliminate the cranks and give a little more chair room.

I would not do a built in bench with a radiator - though you could do one and make the opening underneath a grate - the problem is it will always be the hot seat!

you could a lovely bench with legs and run it parallel to that wall

Because you use round tables in breakfast nooks. They're more casual and they fit better in a smaller square space.