It curbs international tourism, which can help with over tourism. However, it happens at the national or EU level, which makes the impacts of the border restriction far more reaching than other policies.

It also doesn't help if there is a large domestic tourism industry. An EU tourism visa could help filter out some American and British travelers, but you'd still have the Schengen Area tourists to deal with.

Visa free programms are not just a way to get in more tourists. They are often part of international treaties (which often contain way more things with bigger importance than overtourism) or bilateral agreements, which guarantee the same thing in the other country. If a country limits the entry of citizens from another country, that country probably will alao limit down its own visa free programm for citizens of the first country. The countries suffering from overtourism often don't have the treaty freedom to so this or don't want to limit their own citizens travels.

So Visa agreements are mostly part of international politics. Changing that does change a lot more than just tourism, which is often not worth it.

The short answer to your question is yes.

Tourism brings money into the host country. The people who care about over tourism aren't the same people who control visas.

Probably, but I'd just charge more. More people than Venice wants? Crank up the entry fee. Venice benefits and tourism decreases.

This depends on the location. Within the Schengen zone this would accomplish very little. For other countries it might work, but it would probably be easier to just charge some money for a visa on arrival. Extra costs will make some people stay away and it's a lot less hassle for everyone involved. Also, tourist tax is a thing and could be used better in a lot of places.

Politicians don't care for people in the country suffering. They care about the lobbies opinion and the rising BIP. Which both are happy with free travel. Touriat bring in the money (making the tourism lobby happy and the BIP gets a bumb, so they can say:" see I did bring in money. Ohhh the people living near tourism attractions can't affort living anymore? Who cares. Don't you see the money I brought to the coumtry?"

If you want to curb tourism, the only thing you have to do is going after airbnb and the like. And heavily tax (like exponentially tax) each house someone owns after the second. Tourism wasn't a problem when we only had hotels and some eventual Bead and Breakfasts. Tourism became a problem when suddenly everyone could have a room to rent for brief periods.