Does distributor_admin need sysadmin?
In theory no, in practice yes.

After testing for several months on an instance running SQL 2014 Enterprise edition, against 20+ replication instances running MySQL, I found:

distributor_admin does not need SA for day to day operations.

If there is a sync issue (like reboot during patching) if distributor_admin does not have SA it will not self recover.  When distributor_admin does have SA, replication has always self recovered after patching.