[throwawayffffas]

Yes and no, there are cases that you should not use FKs mostly for things like audit logs, order logs, stuff you never want deleted or modified in general.

In general it depends on whether you intend to use the database to drive a system, or whether you intend to use the data for reporting and data analysis.

Inconvenience in migrations is not a legit reason, if your migration would fail if you had FKs but doesn't because you don't you just broke your data, the errors are there to protect you. In other words your colleague is straight up wrong.