I think there's a typo. Should read 'cut the stub further' meaning remove the thick branch altogether. That can work well. You'll usually get some more buds from the trunk nearby that you can grow a new branch.
In this case there are already some existing side shoots in good position so the cut you've proposed could just work. I haven't done it but theory is sound I think. Maybe try it out and see what happens. You can always fall back to total removal if it doesn't look good in a year or 2 and the lower part of the final chop will already be healed.
I'm guessing from the short internodes and graft line this is a dwarf form of JM?