The main problem is that:
0. test/Makefile.in sets HOME=/nonexistent
1. lisp/emacs-lisp/ert-x.el sets HOME=/tmp :(
2. eglot--call-with-fixture tries to detect when HOME is nonexistent,
   but /tmp exists, so that's left unchanged
3. The Rust tools look under HOME=/tmp for which toolchain to use, but
   the answer is under ~USER, so they give up

The following patch temporarily sets HOME=~USER in only those tests that
need it.  The patch consolidates some let-bindings into a single form to
minimise collateral indentation changes.  It also bumps the
rust-analyzer client/registerCapability timeout from 1 second to 3,
because sometimes 1 second was too short a wait.  Finally, it updates
the expected autopep8 reformatting results to what I see locally, and
skips a test that fails without YASnippet.