How do we identify service shutdown time in MySQL
There are three(3) ways using file timestamps:

InnoDB Buffer Pool

If you have innodb_buffer_pool_dump_at_shutdown configured, look for the timestamp of the file that was written. The default filename is ib_buffer_pool. Mentioned this back on Dec 06, 2017 (Control InnoDB buffer pool allocation in MySQL (5.7+) ib_buffer_pool is usually written in folder set by datadir, you can run the following after shutdown:

cd /var/lib/mysql
ls -l --time-style=+"%Y-%m-%d %H:%M:%S" ib_buffer_pool | awk '{print $6,$7}'


Binary Logging

If mysql has binary logging enabled, get the timestamp of the last binary log after shutdown.

Error Log

You are probably saying, "I DON'T WANT TO LOOK INSIDE THE ERROR LOG !!!"

You don't have to. Just get the timestamp of the error log after the shutdown is complete.