For the solution, I reinstall it completely with
yum remove -y mysql mysql-server
rm -rf /var/lib/mysql/ #to make sure it completely deleted
yum install mysql mysql-server
service mysqld start
It works like a charm. Not recommending this solution if the database have useful data inside.
That is a gynormous upgrade.
Plan A: Oracle recommends doing it in 4 steps 5.1 -> 5.5 -> 5.6 -> 5.7 -> 8.0. And, at each step, run mysql_upgrade.
Plan B: Alternatively, you could dump the data and hope that the reload doesn't find some syntax errors.
I fear that you may have already hosed the files too much to have a successful upgrade.