It's well worth reading Samuel Wolfenden's book, "The Art of Pianoforte
Construction", written in 1916 with a supplement in 1927.  He covers
this subject with great clarity.

Wolfenden was chief technician for Aeolian in the UK and is still
widely quoted in piano technology writing today.  His description is
credible and explains some of the effects described in earlier posts.
Not having the book to hand, I hope I remember it right...

Consider the need to get the dead end of the string (between hitch pin
and bridge) up to tension, and the difficulties caused by bad design.
The bridge pins cause resistance in the wire, preventing the whole
string being at the same tension.  Typically the wire behind the bridge
will be 15 pounds lower in tension in a well-designed piano, far worse
in poorer designs.  (There is likewise higher tension in the section
between tuning pin and the agraffe/pressure bar.)

A restringer needs to get the dead section's tension to the equilibrium
level, otherwise the string will slowly slip across the bridge during
playing and the piano will continue to go flat.  Wolfenden says that's
the reason for the traditional wheel: it's not 'stretching' the
speaking length but drawing the wire across the bridge.  He reckons of
some inept designs that it's a miracle they can ever achieve stable
tuning, so great is their resistance to the string.  UK Weber pianos
use graphite to lubricate the string path, clearly part of Wolfenden's
desire to have the lowest string resistance.

Pulling the string a little high to start with clearly will raise the
dead section's tension faster.  It's well within the wire's elastic
limit so quite safe (Wolfenden lists the safe additional tension for
each wire gauge).  However, wire develops a kink where it goes under
the agraffe or pressure bar at the start of the speaking length.  If
this kink is allowed to form above pitch and the tension is then
dropped the kink moves into the speaking length, with a real risk of
causing false beats by not cleanly terminating the wire.

If using a string-lifter device to level the strings and settle the
wire by inducing this kink, it must be done with the piano at its
correct pitch.  The pragmatic advice seems to be that higher tension is
okay for a short while (an hour or two) but should not be maintained.

Wolfenden's book was reprinted by Unwins in a single volume in the
1970s and can be found secondhand for about UKP 10 (or new in some
piano supplier's lists), or the pair of originals turn up for about
UKP 30.  Well worth getting if you are interested in the subject.

Julian Dyer