Previous attempt to fix flakiness in commit 6a2c73f8 ("test: fix
flakiness in gh_6128_background_mode_test") used a constant buffer size
in check_err_msg function. Tarantool 2.10 has a bit bigger log before a
desired message that other versions of Tarantool and it leads to a this
resulted in a truncated message ("entering the even" instead of
"entering the event loop"). Patch replaces check_err_msg()
implementation to grep_log used in luatest, it reads the whole log.

Also patch renames check_err_msg to check_msg, because "entering the
event loop" is not an error message.

Follows up #6128

NO_CHANGELOG=fix test
NO_DOC=fix test

(cherry picked from commit 1c8e7124)