It seems xtrabackup is not backing up binary logs, although it creates xtrabackup_binlog* with apparently correct information inside. Should the binary logs be copied manually beforehand or is this a bug?
xtrabackup the binary logs aren't copied per file (but in one binlogfile).
You need to restore this differently, see docs: Preparing a Full Backup with innobackupex.
Or, with more extensive explanations:
In relation to your database not starting after restore, you might want to paste your
my.cnf and server log