This doesn't make much sense, or there is some info missing.
I" has a hidden danger to it, as there are 1" JIS and 1" ISO, and I think the 1" JIS is something like 0.2 mm smaller than the 1" ISO.
So the early part of headset not fitting and needing to ream the frame could make sense.
OTOH reaming such a small amount is quite easy and should be well within the margins of what the frame can take. Then he says that the HS won't fit even after reaming - and here's where it starts to get odd.
Did the mechanic set the tool wrong, or is he claiming that there was something wrong with the frame to begin with that made the reaming go wrong?

Is he perhaps talking about how deep the the headset cups must be pressed into the frame?

And exactly where is that millimeter gap that might make it hang up while turning?

If what he attempted to do was a JIS to ISO conversion then the old headset should be possible to reassemble after some careful shimming. If that's where the 1 mm gap is other techniques are probably required.