It sounds like you  have a problem with the accelerator and/or the
governor.  This is a simple mechanical slide valve in most players
(pneumatic in a few) that directs full suction to the air motor
when in Rewind.

It could also be that the governor spring is broken, missing or
mis-adjusted.  If the spring has too little tension the governor
pneumatic will close when in Play position, choking off suction to
the air motor.  On some players you can slightly shift the lever
into Rewind, partially opening up the accelerator port and causing
the wind motor to run without shutting off stack suction (sort of).

If the piano is in Play, look at the governor pneumatic with
suction applied.  If it is collapsed there is not enough spring
tension.  Gently pull the pneumatic open with your hand; as you do so,
the wind motor should start to move.  If not, you will have
to disassemble it and check the slide valve for movement and leakage.
If the pneumatic has no suction at all, this indicates the governor
unit is not going into "Play" mode.

The Tempo lever in most players controls another slide valve in the
governor, and the Tempo indicator.  The slide valve may be stuck,
or it slid off its wire due to a rotten leather nut, or the orifice
may be clogged with dirt.

It could also be that the linkage is not pulling the governor rod fully
into Play position.  If this is not the case, then it is a linkage
problem, or possibly a loss of signal if it is of the pneumatic type.
The fact that you stated that both players suffer from this, indicates
that you may have made a similar mistake in this area.

Hope this helps,
Andy Taylor