How will that tax result in more houses existing?

It will result in fewer second homes existing, therefore more first homes.

How will this tax result in less houses available for people with no primary home existing?

It’s a tax on second homes. If a single person sells due to the tax and someone who had 0 homes previously, it’s spread the scarce housing stock.

Not to say we shouldn’t build more but acting like a tax on multiple homes is going to make houses less available to everyone is ludicrous.