It would depend on would make you happy. If they are only replacing some of the windows it may look odd to have some full frame and some inserts and some windows are not the best candidates for inserts.More details and pictures would help in answering the question more thoroughly.

From the information that you have provided, it's sounds like you have existing wood windows. If so and there is no wood damage, a replacement window would be fine. If, however, there is a lot of wood damage then a full tear out would be more suitable. Those are the general parameters of replacement versus tear outs.

The existing windows are aluminum clad. We will be replacing with the same. There is no wood damage. The contractor we're working with insists that the flange and drip cap are integral components for water tightness, and that a window installed without a flange isn't as water tight.

I don't have pictures because the windows are all boarded up.

A window can be installed 100% water tight with or without a flange when done properly, I would not worry about that (or maybe you should if your contractor does not know how to do it).
The one thing that I would say though, is that if the exisiting unit is a full frame unit, the insurance should pay for the same. I'd only accept a replacement/insert as sufficient coverage if that is what you already have.

+1

Windows were done for hundreds of years without flanges and probably leaked less.

Same thing can be said of Tyvek. No matter the band-aid, there is no substitute for proper technique.