Does a ship that's doesn't shoot in the engagement phase still count as engaging? For the purposes of Disabled Power Regulator critical card?

Yes. Attacking is part of engaging. From the rules reference, page 10 (emphasis added) :

The Engagement Phase is the fourth phase of the round. During this phase, each ship engages , one at a time, starting with the ship with the highest initiative and continues in descending order.
When a ship engages, it may perform an attack.

Thank you

Even ships which cannot perform attacks engage, they just don't attack.