This is the wrong way to frame this issue. A lot of cities like Paris and Amsterdam have this issue with short-term renting. "Build moar" is just not really an option for these type of city.
Firstly, the constructible area of the city is limited. So build, but build where ? You can expand horizontally, but this creates challenges with public transportation and other public services. And it can be slow since it means having to expel industry and agriculture further to rezone area into constructible home/office area. So the other option is to build up, which means destroying potentially historic building, changing the skyline and viewpoint. This would be bad for tourism (and people who live here might not like it either), since this is a big reason why people even come to visit.
In the past, cities had simple way to deal with this. With zoning and hotel licences, the city could have a real urban plan on how it wanted to evolve and how much space it wanted to dedicate to tourism vs industry vs offices vs homes. But AirBnb came and just said "fuck that" and bypass complitely the licence système and or building and operating permit usually needed for tourism. Greed and capitalism took advantage of that and the number of place to rent or buy descreased significantly in favour of short term tourism rental, making living in the city slowly unaffordable.
Building more is not that simple. AirBnb respecting the law is a simple solution. It won't complitely solve the issue of the availaibility of affordable home, but it sures as hell help.