People who know me won't be surprised that I have opinions about pizza :-) However, one of the things that I think is important to understand is that different people have very different expectations about what pizza is. I'll try to explain the reasoning for my opinion. You may very reasonably not share that opinion.
For me, pizza is bread. It's bread with toppings, but it's fundamentally bread. I think a pizza crust should define the pizza and the toppings should support that. Because of that initial stance, I believe that toppings on pizza should not be overly wet. Tomatoes should be drained thoroughly before making a sauce, otherwise the crust gets soggy. Any vegetables should generally be cooked first so that the water is removed from them and *then* baked on the pizza.
Traditional Italian mozzarella is a high moisture cheese. It is very different than the "low moisture mozzarella" that is usually found on American pizza. In fact, personally, while "low moisture mozzarella" is a type of pasta filata cheese, I don't think it is mozzarella at all. There a *many* aged, low moisture pasta filata cheeses that are *much* closer to American pizza cheese than mozzarella.
So... it would seem that I should prefer American pizza cheese on pizza. But I don't. To me, the secret is that mozzarella should be added in very small quantities. While the soft stretchy texture is nice, IMHO, the reason for putting mozzarella on a pizza is the milk fat. In Frace, it is common to put creme fraiche on pizza (ham and raw whole eggs, baked and then drizzle with creme fraiche... Has to be tried!) It's that milk fat coupled with the lactic acid from the ferment (and this is why you need actual traditional mozzarella and not "quick" mozzarella that uses citric acid). With an Italian pizza you then drizzle a little bit of olive oil on the top. That mixture of milk fat and olive oil is heaven. However, you need to be conservative because otherwise you just get a soggy mess.
There are lots of cheeses that will fit the bill for that kind of vision. The main thing is a fresh cheese that melts enough to allow the fat to ooze out of it. You've got to be careful of moisture content -- the more moisture it has, the less you can use. A cheese from milk with high milk solids is going to be easiest to use -- water buffalo milk and sheep's milk are going to be the best, but some breeds of goats and cows produce milk with very high fat content. You can also make cheese with added cream. I wouldn't personally use aged cheeses most of the time because you want that fresh cream flavour, not the flavour from the action of lypases or proteases.
Having said all that, you can make amazing creations with all sorts of ingredients. I love pizza with feta which is pretty much the opposite of what I'm suggesting. I've made some really nice pizzas with aged gouda cheese. Blue cheese pizza is amazing. In fact, a very common pizza in fancy restaurants in Japan is gorgonzola and honey. It's amazing. I've also made pizzas with Brie or Camembert and honey. Ground beef oinions and cheddar is very nice! One crazy pizza that was popular where I grew up in Canada was: bacon, onions, potatoes, cheddar cheese and sour cream. If you have a cheese, I think you can make a good pizza with it. And (horrors), I've had *many* great pizzas without cheese. Historically pizza without cheese was very popular! Just stay away from the popular-in-Japan: mayonaise, fresh corn, potato combination (somehow made even worse by the offer to crack a raw egg on it when served at your table).