Ok, i really dont see the one inch gap under the eve metal being a problem that cant be overcome.
The drip edge is the appropriate size.
and it is appropriate to cut around any gutter fasteners.
Bugs want to go where it is not hot
The gutter looks pretty tight on the bottem against the fascia.
So your goal would be to get the eve metal flush
Against the gutter
And may require extra fasteners through the face as you have already done to achieve it.
That is what i would do.
Or if you must just have perfect peace of mind and do something different
You could roll out a screen in front of it
Before you install the new eve metal.
Or you could install a 1x2. A true 1x2.
Most 1x2 is only 1 1/2.
Not big enough to close the gap and have something to nail to.
Your pic does reveal something that is worse than the issues you think you have.
When you get your new roof, make sure you tell your roofers to NOT cut the modified even with the eve metal.
It is what almost all of them do.
It is what i did early in my career
It looks the best and is water tight for many years.
But when it gets older, it fails right in line with its warranty- 12 years or sooner.
The material shrinks back away from the metal
(And any side seams in the field)and water rots out your decking/fascia.
So to rememedy this, After all underlayment is layed, i make sure the edge metal is sandwiched well between the new material And the material overhangs the eve metal and “rakes” especially at least a half an inch or more.
It doesnt look as nice
But is mechanically superior and can ensure the roof lasting 20-25 years or in definitely with coatings.
Double its warranty.
I am fairly confident that the roof is leaking at the top leading edge where the material has shrunk back from the metal. The seal is completely broken.