'So the approach I suggested is to find a set of end-game positions where
the outcome is always a draw or always a win, with extreme positions
excepted. This might produce different values.'

The problem is that such positions do not exist, except for some very
sterile end-games. In theory every position is either a draw or has a
well-defined 'Distance To Mate' for black or white, assuming perfect
play from both sides. But this is only helpful for end-games with little
enough material that tablebases can be constructed (currently upto 6 men
on 8x8). And even there it would not be very helpful, as the outcome
usually depends more on exact position than on the material present, and
thus cannot be translated to piece values. For a given end-game with
multiple pieces, there usually are both non-trivial wins for black as well
as white. (Look for instance at the King + Man vs King + Man end-game,
which should be balanced by symmetry from a material point of view. Yet, a
large fraction of the non-tactical initial positions, i.e. where neither
side can gain the other's Man within 10 moves, are won for one side or
the other.)

And for positions with 7 men or more, we don't even know the theoretical
game result for any of the positions. This is why Chess is more
interesting than Tic Tac Toe. It is impossible to determine if the
position is won or lost, and if you have engines play the same position
many times, they will sometimes win, sometimes lose, and sometimes draw.
So the only way to define a balanced position is that it should be won as
often as it is lost, in a statistical sense (i.e. the probability to win
or lose it should be equal). Positions that are certain draws simply do
not exist, unless you have things like KBK or take an initial setup where
a side with extreme material disadvantage has a perpetual.