test: fix flakiness in gh_6128_background_mode_test

Test runs an external process with tarantool that writes to a log file.
Then test reads that log file and searches a string with required
message in it (see function check_err_msg). Test was flaky on macOS and
I suspect it was happening due to a high log level - timeout was not
enough to wait message in the log file.

Patch decreases a log level to a default value and replaces io
functions with the similar alternatives in a fio module. Using
fio functions allows to not block fibers.

NO_CHANGELOG=test fix
NO_DOC=test fix

(cherry picked from commit 47380bb7)