I would say that it is not the hole that makes them bagels, but the boiling of the dough before the cooking.

After all - if it was the hole that defined bagel-ness then ring doughnuts would be bagels... The hole seems to be to ensure even baking - so if you took the dough, boiled it, and then baked it in a different shape it would be a different shaped bagel.

But then I am not a Jewish baker, so what would I know?

Apart from the fact that soda bread is much nicer if you want a change from ordinary bread rolls...