As far as I'm aware, you can't utilize a gutter box like an extension ring, although the consensus thus far seems to be that you can, interestingly enough.
I would see that as a violation of 110.3(B) personally. However, it opens things up to the debate about being able to fabricate in the field, like fabricating your own knockouts, which many on here seem to support, and rightfully so, IMO. How else are you supposed to deal with boxes that come with no KO's? I just don't think turning a gutter box into an extension ring is anywhere near in the same category.
A quick look at your picture and it appears you have 1 x 4" SQ 1900 w/ 1 x 4" SQ 1900 Extension Ring = 42 CU.
I count 18 x #14 + 1 EGC in NM which equals 38 CU / 42 CU, not counting what comes in from the EMT.
That means nippling over and transferring all those existing NM's is NOT physically possible.
You can't do you're first proposal of extending to the surface because the existing wires will not reach, what I believe is a required 4" past the front edge of the box, but I could be wrong about that. I was always taught 6" of lead in the box, 4" past the front.
So, assuming that we agree that neither of your 2 proposed solutions are acceptable, I would consider a solution that involves rerouting everything to either one of the following:
(1) 1 x surface mount 6 gang.
(2) 2 x flush or surface mount 3 gangs
(3) 1 or More j-boxes higher up and then piping down to your cleaner, preferred solution and bite the bullet on the blank covers above.
All of this is assuming this closet will no longer serve as anything but for electrical.
It appears to me that shifting up, rerouting all the existing cable, if possible, and just having to reroute one EMT is the best solution, IMO.