I can only advise you on what I think you will find.
You already mentioned the best possible way to start correcting the situation, remove the gutter and finished fascia, but you cannot stop there, you will need to remove the top piece of siding that meets the soffit as well. My thought is, the metal soffit is very thin, and the soffit that you neighbor has, is standard soffit material that is available perforated to give the ventilation you need. It is 5/8" thick, so it will not fit in the same space the thinner original soffit does. The nails holding the last piece should visible at the very top, just under the trim strip.
When you remove all this you will have access to the sub fascia to check its condition. The ice forming behind the gutter, is not a good thing, water may be getting behind the metal and rotting the sub fascia, so it is worth a check. On your neighbors picture, the ice appears to be coming over the front edge only, where it should be. Since you should be going with the thicker soffit material. you will need to remove the track at the wall too. The soffit will be a little lower since it will contain the 5/8" soffit, and any damage you create on the siding from its removal can be cut off. After all the soffit is installed new, set a new, or possibly reuse the original "J strip" at the top hide the edge of new soffit at the wall and to capture the top edge of the siding after it is trimmed off to its new dimension. Set in place with new nails like the originals, pilot first. The old hole may have been cut off by the needed trimming, or at least hidden better by the "J strip".
That piece at the top that caps your siding now may be a "trim strip" very thin, maybe 3/8" thick at the most. If that is the case, you would do well to add 2 "J strips" or an inside corner specifically made to handle the soffit and siding going into it from both directions. You may be able to do this with one J strip, add it after the siding is on WITHOUT trimming it as mentioned before. Slot the nail hole to allow expansion and contraction. Set the j strip to the rough soffit to hide the top of the siding, set the soffit material and set the fascia and gutter last. There is a small chance the last piece has been "dimpled", and is held in place by locking into the trim piece, Ease out the trim piece edge and it MAY come out. If it is caulked in place, it will be a bear to get out.
You may need to get a taller piece of fascia to accommodate the 5/8" taller face you just created, it may need to be taller any from the symptoms where the ice is. If not the gutter may have not been installed so the front edge is HIGHER than the back, it is not supposed to be. Typically the back is high enough to put the back edge, IF possible under the drip edge so any runoff of the roof is forced into the gutter.
If you want to consider an alternative that you may retrofit easier and may look better too, is this line of continuous soffit vent. This link is for a full box, but you may find a smaller quantity elsewhere. It is made to set in with a thin plywood soffit, but the edges can be cut off which would make it quite flimsy, or you could finish making the bend that would "hem" the edge making it a bit stronger. Razor knife the section at the wall, remove the soffit and gutter, which I think would be a good move to inspect the framing. Pull the nails if any holding it in place at the fascia, work the old soffit so the score mark you made at the wall will flex and the old piece will come down. Take care not to distort the edge you cut, bends here will make trouble trying to get the new piece in. Slip this stuff in under the cut at the wall, between the old soffit and trim strip for the siding and tack it back in place before you reset the fascia.
If you have any small pieces of drywall up on the ceiling, I'd start by removing those to get back to as many full sheets as you can which should also minimize the cuts you need to make. As user41750 says, make sure you match the drywall thickness. The joints will only be staggered in one direction, along the 4' lengths where the drywall isn't tapered. You want those 4' long joints to meet on a joist or nailing surface.
The 8' spans will be perpendicular to the joists and be tapered. You can knock out the mud and tape from inside the taper of the existing drywall to get two tapered joints to align. If you don't have an existing tapered joint, then you'll want to cut away the taper to avoid a significant difference in height where you are mudding and taping later.
The mudding and taping is an art. There's a fair bit of advice on this site to help you there, but I wouldn't blame anyone for hiring out this part of the project. And from there, it's all about matching the finish. I'd personally prime and paint the entire ceiling, if not the walls too. This will help hide any division between the two spaces.
Best Answer
To be perfectly honest with you it may actually be easier and less overall work to remove the rest of the kitchen ceiling and redo the whole thing with new drywall.
This would be especially true in the case where the kitchen ceiling is enclosed by walls or other partitions. If the ceiling merges into the ceiling in another room then the suggested process may be a bit more work. I still think though that a merge patch in the ceiling between two rooms may be less work to make look right than the type of patch work required to fixup just where the cabinet soffits were located.