mysql xtrabackup missing binary logs
When using 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:


Percona documentation - Prepare with xtrabackup
Percona documentation - Restore with xtrabackup


In relation to your database not starting after restore, you might want to paste your my.cnf and server log hostname.err.