Have attached a photo of the new roof here for comparison. .

If the heat gain is low there is no need for a radiant barrier. If the eaves can be repaired there is no need for a new roof. Perhaps wood can be installed between the rafters to hide the new foil.

seems like a lot of cost & work to remove the whole roof
of original part of house.
I'd replace @ eaves...and find
out & solve why this area rotted..take steps to keep it from
happening again.

then install a radiant barrier attached with
foil facing to attic space to the undersides of the rafters.

I use button cap nails rather than staples & recommend the
least expensive radiant barrier product as they all perform the same.
mine is a fsk (foil scrim kraft paper) with foil on
only one side. performs the same as foil bubble foil for a
fraction of the cost

here is a pic of my install.
had a little repair to do after katrina & rita
passed my way!

best of luck.

Are the rafters visible from below or not?

Thanks for the replies, the rafters will not be visible since we will be putting drywall ( I guess) to create a flat ceiling. Sorry dont know the construction terms. Since the eves were rotten from exposure to elements, we had the entire roof replacement as part of the contract, but now I like the idea if having more wood rather than just a thin piece of plywood ( radiant barrier) and replacing the just the damaged part.

That's helpful to know. IMO what you need between the rafters is insulation. Roof ventilation is another matter but you haven't said if there are any roof vents.