Why can the relative pronoun be omitted?

Solution 1:

[1a] My brother, who is a former professional hokey player, coaches elite high school hockey.

[1b] My brother, a former professional hokey player, coaches elite high school hockey.

[2a] Teachers who work online have flexible schedules.

[2b] Teachers working online have flexible schedules.

[3a] The man that was bitten by the dog went to hospital.

[3b] The man bitten by the dog went to hospital.

You're not omitting the relative pronoun, but replacing the relative clause with a noun phrase or a different kind of clause:

In [1b] with an appositive noun phrase, in [2b] with a gerund-participial clause and in [3b] with a past-participial clause.