In the vast majority of cases a #2 or #3 tip will fit the tuning
pins well.  I find that not all #2 and #3 tips are alike, which is
why I carry a big selection.  Because you did try so many different
tips, I think your problem is most likely the previous tuners of this
piano over the many decades.  Tuning pins wear the more they are
turned, and squared edges become rounded.

I was taught that the motion of the tuning lever should be (more or
less) parallel with the pinblock (compensating for the angle of the
tuning pins).  Many tuners unfortunately do not grasp this concept.
These tuners turn pins with poorly fitting tips and often will slap
the tuning lever (with a big running start) or jerk it in a smooth way
from all kinds of angles.

You'll just have to work with the situation you have.  I often have
to "make do" with a tip that does not fit well.

Bill Maguire