As I mentioned in the article, the old ID field was then used in the update statement for the other tables.
I'm updating the article now to make that part clear.
But even that aside, I still say the binlog_format is irrelevant and the core problem here is 100% the ALTER to add the auto_increment: it resulted in different IDs on the replica than on the primary. That's a problem if you refer to IDs externally anywhere, regardless of whether it's 6 child tables or an external cache or logging etc. As soon as you promote a replica for any reason (not just an upgrade, any failover reason whatsoever) this would be a massive problem as all the IDs would now refer to different rows. And even before a failover event, if you do any reads from the replica for any purpose (read scaling, backups, OLAP queries), the data is going to be wrong.
Essentially for the 6 child tables, it wouldn't have mattered if their UPDATEs had all used ROW or all used STATEMENT; either way you would have still had a fundamental data inconsistency between primary and replica here for the parent table. If these 6 tables' UPDATEs all used ROW, they would refer to the IDs from the primary which are locally "wrong" on the replica. Or if they all used STATEMENT then the data on the replica would be consistent locally but completely different than what's on the primary.