That is somewhat true but what this also causes is that poorer people have a harder time travelling and visiting other countries while rich people are unnaffected
The common good and environment of a nation has priority over cheap tourism, so I don't necessarily see that as a problem. Especially when the number of annual tourists exceeds the population by a factor of five in the case of Iceland, I can understand why some residents would like to reduce it.
I'm sure the entire government of Iceland, including any departments dedicated specifically to, and with decades of experience in tourism, could figure it out if they really wanted. But just charging a tax is easier, so here we are.