Hard to really tell cuz none of us here can actually hear what your bike is doing but I'll take a shot. There's a lot of great techs here so, hopefully we can help ya trace it.
I'm not leaning towards an electrical issue. Though not impossible, seeing as it idles fine and revs better (but not good) in neutral but not as well once under a load (in gear) I'm still going to lean towards carbs.
Sounds like you may not be getting enough fuel.
You say you cleaned them but, a real cleaning should involve pulling the jets and tubes and really cleaning them out. Take a small wire, an e string from a guitar works great, and poke it through all the holes in the tube gently. do this until you can put the little hose that comes with a can of carb cleaner, spray it and see a steady spray coming out of all the holes. Make sure you spray through all ports and clean the bowls completely. Any little bit of schmeg can clog up a jet. Next, after you assemble them and get them back on work on your pilot screw adjustment. This puts more or less fuel in your crabs. I believe stock on these are 3/4 turns out from gently seated. Do this, clean your plugs then take it for a quick ride. Then check your plugs again. If they are white you're running a little lean. Turn the screw out a little. If excessively black you're a little rich. Turn them in a bit. You'll have to adjust each carb to how the coinciding plug looks. Of course, getting them properly sinced will help a lot too.
Try this and let us know how it turns out.
One step at a time my friend. But no, it doesn't sound like you'll have to go as far as pulling the whole motor out. :)
Good luck
It ain't a custom till you have customized it yourself.
1981 CB750c (current daily ride and build)
1980 GL500 (Stripped and rebuilding)
1981 CM450 (Stripped and rebuilding)