The only cheese I've made with just curdling pasteurized milk with acid is paneer. Which is basically a ricotta, except a lot of ricotta recipes also use cream along with the milk.
I did press the curds to form a block or chunk and it was sort of sliceable. It was OK. I added some herbs, etc. as well. I was playing around with ways to salvage milk that was getting long in the tooth. I haven't even thought about making it since! Not sure why...
Since I learned how to extend the fridge life of milk, I haven't had the dilemma. The key is to reduce the amount of air in a container/jar of milk by downsizing as you consume it. I would buy a gallon and as soon as I opened it, I would transfer 3 quarts in to quart mason jars filled to the top and used plastic caps rather than the canning lids. As I used each quart, as soon as what was left it would fit, I would transfer the remaining milk to a pint jar and so on, to a half pint jar. I would make a gallon of milk last far beyond a month and that last quart tasted just like the gallon when I first opened it.
I've always been interested in making my own cheese. I just got real busy with other stuff and forgot all about it. Since going low carb diet I never drink milk anymore but cheese is totally allowed on the diet. I should get back to looking at making my own mozzarella.