The rule you cite is mostly true, but it's not complete, because just mentioning "relative pronouns" doesn't distinguish between
wh-words like who and which, on the one hand, which can occur with all relative clauses,
and
that, which can occur only on restrictive (integrated, defining) relative clauses,
- The man whom we met yesterday was a sales representative.
- The man who we met yesterday was a sales representative.
- The man that we met yesterday was a sales representative.
- The man Ø we met yesterday was a sales representative.
But that can't occur on non-restrictive (supplementary, non-defining) relative clauses,
(in what follows, ungrammatical English sentences are *marked with asterisks)
- The chairman, who(m) we met yesterday, was once a sales representative.
- The chairman, who we met yesterday, was once a sales representative.
- *The chairman, that we met yesterday, was once a sales representative.
- *The chairman, Ø we met yesterday, was once a sales representative.
Non-restrictive relative clauses also bar relative pronoun deletion,
as the last ungrammatical sentence shows.
The rule also doesn't distinguish between most relative pronouns and those relative pronouns -- either that or wh-word --
that are the subject of their relative clause. Subject relative pronouns are also immune to deletion; English tensed clauses require a subject, and this requirement overrides.
- The man who met us yesterday was a sales representative.
- The man that met us yesterday was a sales representative.
- *The man Ø met us yesterday was a sales representative.