Your desired outcome is to compare both offers and take the one you like best. So yes, I would mention to the second company “I have an offer in hand from another company. I’ve really liked our discussions, however, and was wondering when I might hear about an offer from you.” They will then accelerate as possible/desired on their end. I would then also put off the first company a bit, saying “I will need a little more time to decide and won’t be able to join on that aggressive of a timeframe.” Ideally you get the second offer before the first one gets impatient, and you can make a decision.
Of course it depends on your situation. If you’re unemployed and desperately need a job, then it’s not a good idea to do this because you could be left with no offers. But for a normal job change without undue pressure, some brinksmanship is fine.
Is the slowness a “red flag?” In general I’m not a fan of everyone deciding anything a company does inefficiently is some kind of deal-killer; that’s just Internet Tough Guy talk. But it is a factor to take into account as the overall package of which position to take.
I had a very similar thing happen to me - I’d been talking to one company about a leadership position and they started going through the same gyrations, but for months. “Well the hiring manager left...” “Well were having trouble with a perm role, how about contract to hire? No?” (I was in a perm job with benefits, and their role was not so stunning that I wanted to do my own benefits and have greater risk of he job being short term, etc.) It took more than 3 months for them to finally sack up and put together an offer. But of course just as they did, I got an offer from a smaller company where I knew someone I’d be working with. I took the second position based on a mix of factors - the work, the people, the pay, and, yes, because they had revealed to me through the process that probably as a leader in org #1 it would take me an untoward amount of time and effort to hire (and possibly to do other things) as well. Company #1 was upset because of the “time and effort” they’d put in over months of trying to get things to work, but of course so had I.
But having said that, I wouldn’t have accepted a radically worse offer from company #2 just because company #1 had sand in their drawers; if they were offering lots more money or higher position or more interesting work or something I’d “get over it.” They weren’t, and one of my goals was to build my own team so their hiring travails were more than just an arbitrary “bad thing” to me but would directly affect my ability to do what I needed. But I wanted to change jobs because the company I was with sucked (had been bought out by vulture capitalists), so I would have taken offer 1 if it was the only one.
That’s a long anecdotey way of saying don’t be suckered in by people trying to get you to make overly simplistic career choices based on one specific detail. Work to optimize your own career; make smart decisions and tradeoffs about when and how hard to push for multiple offers, factor in all the attributes of offers when deciding on them, and so on.