I think you're very misinformed and have been hearing a lot of sob stories that are likely 95% bullshit.
Have you ever had a long-term rental(s) next to or across from you that was purchased by a "slum-lord" type property management company, or better yet, a wealthy parent, for their kid in college and all their partying buddies and friends/vehicles? I can tell you that 100% of the STRs I'm familiar with are appreciated by their neighbors, far more so, than I have been with terrible long term renters that I've had near me a few times. Night and Day.
STRs are occupied by loud, destructive, partying vacationers? What evidence do you have of this? Do you know how VRBOs work (ratings)? They would not be successful over time serving a market like that - for either the renter or tenants. I've NEVER once seen that in 5 years of being involved with a rental and the neighbors. Also, I've never once behaved like that when staying in one.
STRs are being mass purchased by wealthy investors, to largely sit "empty". Only for "tourists" willing to pay through the nose? Sounds like you think options for visitors to Bozeman and elsewhere should only be a $150-$600/night for a bed, shower and a toilet?
Wealthy out-of-staters are "renting" houses and only stay in them part of the year, as a result of the rent going up, due to STRs? The "shortage" in homes and "long term rentals" is a result of STRs. How do you figure that? Got some data?
It sounds to me like you've been talking to folks that didn't work, save, invest, and buy a home, or at the right time, and now looking for something to blame. Whoa is me, I had to pack up and move to Helena. Boo effing hoo.
The only STRs I'm familiar with are owned and maintained by working class locals, who worked, saved and bought a home at the right time. They don't want to rent them long term as they often have to be gutted after tenants move out, there's less control over the condition and upkeep by the owner as they might move back into it someday. There is more income potential, but it takes way more work.
You really think you should "regulate" who gets to purchase a home?
FYI, STRs are "regulated" in Bozeman, licensed, taxed, and only allowed in limited zoning, unless they were "grandfathered" in 3-4 years ago.