Doesn't TeX respect LC_PAPER?
No. TeX/LaTeX is letterpaper by default. You have to ask for a4paper
in the LaTeX source (or, in plain TeX, meddle with \hsize, \vsize and
stuff) explicitly to change the layout. While dvips has a default paper
size setting, that does not concern the dvi file (and consequently the
text layout), but merely the offsets for placing the page content on a
particular printer.
I can understand that Lilypond should not look at any local settings
to interpret the input, as the input is Lilypond-specific. But I
don't see why Lilypond should ignore local setting for the output.
Paper is paper, whether it's used for text or music.
If you are asking for LilyPond to behave like TeX/LaTeX, the result is
exactly _not_ catering to LC_PAPER.