Cap Old Gutters and Insulate?

Moisture has to have a path in to create a problem there.

I agree with your contractor that you don't need ventilating there. House should be loose enough that ambient air moisture should be passing in and out. I'd just make sure that the side wall is flashed correctly to the old gutter area so that there is always physical barrier for rain to wash away from the house trim/siding joints. Looks like the best will be to use xps foam boards there and seal any joint with canned spray foam. Not much insulation value gained for a small area. But that kind of area usually leaks lots of air. Insulation can start under the wall shown, does not have to be all the way out to the ends of the framing. Insulation is enveloping the living space. If downstairs is a room, then out to the end.
If that image room is not drywalled yet, I would look for air sealing joints between the side roof and the inside wall. Fiberglass insulation loses lots of r value if air is leaking all over and through it.

Thanks for the response.
I will get it all insulated/sealed up and will make sure all joints are watertight.

Kind regards
Lumin