There's also the meta level issue that holding site operators responsible for the actions of their users is a very slippery slope. Anything that chips away at the communications decency act could have terrible effects for the internet as we know it[2]. For example, as it currently stands if I were to libelously claim that Paul Graham has a meth lab in his basement[3], I could be charged with a crime, but not HN. If the CDA were weakened, blog operators could be liable for comments posted on them. I don't think anyone reasonable wants that.
I think SF's law did a good job balancing things - hosts need to be licensed by the city (the app lists their license #) and can IIRC only lease 90 days out of the year. I looked at a recent receipt for a stay, and they even collected taxes.
[1] https://www.eff.org/mention/making-civil-liberties-case-airb...
[2] https://www.sfchronicle.com/business/article/Airbnb-HomeAway...
[3] To be clear this is a hyperbolic example, and I have no reason to think PaulG has anything illegal in his basement :)
Another problem is that the proliferation of this practice seems to be disturbing the already tense situation on the housing market. Meeting an odd couple from the other side of the world is fun. Realizing that you can't find affordable flat to buy in a city because a significant portion of it is scooped up for renting and short-term leases - that drives up societal discontent.
And tax evasion is a completely different topic, that is not why Airbnbs are shut down. Every industry has tax problems.