Nimble,
Not read my second post?
There are more than a single scenario with these problems.
A crooked hole – pilot drill wanders and subsequent drill follows.
A hole that is belled at the start due to excess pressure on a flimsy base. Tap not started straight.
A hole that is drilled straight but no longer perpendicular due to a flimsy machine where the drill pressure deviates the hole-angle from perpendicular but the tap is presented perpendicular when tapping.
Simple poor starting by hand – not threading down the line of the hole.
Not all taps, held in a tailstock, are straight – with the tail stock quill extended excessively and seriously worn it can result in a thread being started off-line. Lousy practice, but can happen. The tap only needs to start a thread crooked and it will continue on that line. It will not correct itself, once started crooked.
In this thread starter, the OP did not give any useful procedural info – only that the tap stuck part-way through the hole. Nothing unusual in that – it happens so often.
Later, the suggestions are refuted until an apparent suitable suggestion fits the bill. Surprising that the taper tap was mistaken for a another, after so much use over the years, and not checked/noticed when the problem occured. Might have happened here, might not. An 8.6mm drill might not cut an 8.6mm hole. Even an 8.7mm drill might only cut an 8.6mm hole (was the diameter verified?).
I might have even questioned whether the set of taps were really only thread chasers, but I did not. One threading, with an unused tap from a set that has otherwise apparently been brilliant over the last umpteen years, seems out of place. That the OP had not attempted even a second threading, before posting, was a bit mystifying as that would be my first check, had I had this problem.
Analysing the whole thread just makes me wonder….