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i had a very similar idea a few months ago, and am a little disappointed
to
learn i wasn't the first to think of this. i was inspired by a game
called
russian square, in which you must push coloured blocks around in a square
(or rectangle) until a whole file or rank is one colour. it then
disappears.

one problem with your shrink chess idea that hasn't been addressed here
is that of internal files and ranks. my idea only allows external files
and ranks to disappear. i have 3 reasons for this. first of all, the game
of chess begins with 4 empty internal ranks. under your rules, 3 of them
would disappear after white's first move. secondly, discovered check
would sometimes happen if an internal line disappeared. this is not
neccessarily a good or bad thing, but i don't like it. thirdly, if an
internal line disappears, the checkered colouring of the board would be
violated. it's easy enough to just change the colours on one half of the
board in order to restore order, but nevertheless sometimes this will
result in both the bishops for one (or both) player(s) being on the same
colour square. not great.

apart from 'russian chess' or 'russian square chess' i didn't really
have a satisfactory name for my variation of shrink chess, so you can
call
it shrink chess 2 if you like.
unlike shrink chess (1) a separate castling rule is not needed. only
emptied external lines disappear. pawns only being able to double step
before any rows/lines(?) are removed satisfies me greatly. particularly
since, (in my opinion) the first line to go will do so sometime towards
the end of the middlegame. however, i'm not at all keen on the idea of
white being able to choose the promotion of black's pawns (and vice
versa). however, if my opponent does send my pawn to the last rank, i
could then choose a piece which would check his king, and then take it
with my move. i expect there is a satisfactory compromise here. i just
have not thought what it is yet. maybe you can only choose a piece which
doesn't check the opponents king. also, that gives a bit of the
promotion
power to the opponent, instead of all the power, as in shrink chess (1).
for example, 'my opponent can promote a pawn on their next move, if so
he
will probably choose a queen. however, if i promote it and have my king
in
line with it, i can ensure he only gets a minor piece for it.'
i quite like that.