Intonation can be easily set on an acoustic using a compensated saddle. If there isn't enough space to set it correctly, the saddle slot can be filled and re-cut. The bridge doesn't have to be removed and re-glued. Trust me, i'm having this done as we speak on my AJ-307 with .011 to .070 strings on it.
I have a .060 low string on my Dean 7-string, and i seem to be able to intonate it correctly with its tune-o-
matic bridge. I think i could easily go up to .065 on it, maybe even higher. And those things don't have a lot of play.
I think you may be right in assuming there's a problem with the nut. What nut material is on your 7621? I'd take off the string and see if there's any chipping or flaking at the leading edge of the nut (closest to the fretboard). If the point where the string breaks over the nut has moved back toward the tuners, it would definitely throw off the intonation across the entire neck. I can't think of any other reason why you'd need to set a .065 back THAT far.
If you do have any chipping or cracking, or if the slot was just badly cut, see if you can get it replaced with a good quality GraphTech, Tusq or bone nut and have it slotted by a professional who knows what they're doing.