Let me put it this way: you *can* do different things, but it won't be easier
OK. It's an excuse for my favourite topic: acidity and acid coagulated cheeses. Let's see if I can do it this time in less than 5 million words...
Acid coagulated and rennet coagulated cheeses are different. Rennet works by modifying the protein in milk so that it can all hook up together using calcium as a glue. The gel that it makes is relatively quite strong. You get a rubbery, strong cheese. Also, rennet will work with relatively little acid. In fact, you can make halloumi with no added acid at all -- basically the same acidity as milk.
The proteins in milk are normally "hydrophilic". This means that they love water. They stick close to the water. When you add acid, the proteins stop liking water so much. When you add enough acid, they don't like water at all. This is called "hydrophobic". They race away from the water and kind of clump together. The clumps are weak, though. It's more like sand in a sand castle than the rubbery cheese of rennet cheeses.
There are 3 very important things to understand if you want to make acid coagulated cheeses: First, the faster the water becomes "hydrophobic", the faster it rushes away from the water and the bigger curds it makes. If you add acid very slowly, over a period of several hours, the curds will often be so small that you can't even see them. Even though the proteins are hydrophobic and clumping together, those clumps are super small and so you end up with basically something that flows -- yogurt. If you add all the acid you need very quickly, say 2 seconds, the curds will clump very large. You can get curds that are several cm in size (bigger than an inch) and you will see the curds and whey split.
The next most important thing is that heat damages the proteins in milk. If you bring it up to a high temperature, the whey proteins get all scrambled up (exactly like cooked eggs) and they get tangled up in the protein that we generally use for making cheese: casein. This means that the proteins won't clump as well. The tangled up proteins *also* trap water, which makes your yogurt thicker, but also makes it harder to drain when you are making cheese. The less heating the milk has, the easier it is to make large curds of cheese that drain well. Ideally you should use pasteurised milk (or raw milk if you are OK with the minimal risks involved with that). You *can* make acid coagulated cheese from UHT milk, but it's easier without -- which is the opposite from yogurt.
Finally, the amount of acid you need to coagulate the milk depends on the heat of the milk. Usually we measure acidity with the pH scale. Lower numbers are more acidic than higher numbers. Normally yogurt forms at room temperature at a pH of about 4.8 (the same as feta, blue before you add blue mold or camembert before you add the white mold). If it is 50 C, the pH is more like 5.3 (about the same as mozzarella or cheddar cheese). If it's 85 C, the pH is more like 6.1 (the same as ricotta cheese, because that's how you make it ;-) ).
You can control the acid level of your cheese by heating your milk to a specific temperature and then adding acid until the curds form. Because you add the acid quickly, you get large curds that are easy to drain. Unfortunately, acids that you add to milk do not make very delicious cheese. Also, if you age the cheese, it lacks the enzymes to produce nice flavours. So it will always be lacking. However (and this is the magic), you can go the other way around!
In other words, grow your yogurt bacteria in the milk until you get the acidity that you want and then heat it quickly to form the cheese. Because you are using a fermentation process, you get nice flavours, you get lactic acid instead of citric acid (or acetic acid) *and* you get enzymes from the culture that will eventually produce nice flavours as the cheese ages. Hooray! And because you heat the milk quickly to get to the point where it curdles, you get big(ish) curds that drain easily.
The main thing is practice. You add your culture. You leave it at a specific temperature (depending on the culture) for a certain amount of time, you heat the milk until it breaks, you drain it, you press it lightly, you salt it. You age it. Bob's your uncle. (I don't know if the last bit is true...) But it will take time to get to know your cultures. One word of warning: If you leave your culture so long that it makes yogurt, it has *already* coagulated. it won't coagulate again. So in order to get big curds you want to heat it wile the milk is still milk. (BTW, this is not true for some types of cheese. There is a famous style of brie cheese that is made by draining yogurt, probably in a similar manner to what you are already doing. Edit: I forgot to mention that to get a nice solid cheese from draining yogurt, normally you need to drain it at room temperature for at least 3 days -- sometimes it takes my cheeses as long as a week to get to the consistency I want).
There is much, much, much more to this, but I promised to keep it as short as I could. It's enough to get you started, anyway. There are many traditional acid coagulated cheeses that are aged. Some are soft cheeses, but some are hard cheeses too. They have a different texture and flavour than rennet based cheeses, so don't expect to duplicate cheeses from the store. Having said that, though, the cheeses are *different*, not worse.
I will leave you with one last thing, though. Keep in mind my first words: this is not easier than rennet based cheeses. In many ways it is *harder* to make quality cheese this way. I have to admit that I probably make almost as much acid coagulated cheese as I do rennet coagulated cheese, but I'm crazy. You don't have to be crazy.
