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So what can you put up with? Unless one is willing to have what could be
considered an unwieldy number of pawns [at least], then the double pawn
wall on which so much of modern chess depends no longer exists. A piece in
a 3 or 4D game can move off its starting cell to somewhere near enough to
the middle of the board on the first turn, without any pawns moving. From
that spot, the piece will be able to either attack an enemy piece, not
pawn, or move to another spot from where an enemy piece can be attacked -
if we use infinite sliders. And here, Charles, is where we come back around
to your nicely-conceived idea and the use of short-range pieces as a
partial but fair solution to the problems posed by 'infinite sliders'. I
don't suppose it's surprising that I recommend short range pieces, but
that doesn't make me wrong in this case.

The board will still likely be in a probably unguessable state in just a
few turns, but since the pieces are short range, the chaos of the situation
is limited in scope. Players can see an attack gathering/coming, even if
they can't see just where it might land, giving some chance to react. Face
it, attacks will be far more flexible in higher D, so giving the defense a
little more flexibility gets a bit of the imbalance back.