Web browsers are already running random expressions from the internet, but their regex implementations have proven the rule by being fodder from exploit writers. In any event, requiring a password manager to pull the engine from Firefox or Chrome wouldn't be cool. They're more likely to use libpcre or re2, but see above.
The NAPTR DNS record type uses regular expressions for domain rewriting: https://en.wikipedia.org/wiki/NAPTR_record Not many DNS libraries support that record type as it adds alot of problematic bloat.
Evaluation is linear, but you're just evaluating your own password. In this scenario, it may make sense to just use a simple backtracking implementation, though as you alluded to early there aren't many simple ones around.
I totally agree with your concerns about scope creep increasing the attack surface.