If you were to tax a church, you'd be "prohibiting the free exercise thereof" since taxation is a control, and thus, in violation of the First Amendment.

I'll have to disagree. Freedom of Religion is like Freedom of Speech. It is about what you are allowed to believe, like Freedom of Speech is about what you can and cannot say. Freedom means that you can believe (or say) whatever you want. You could for example believe that all humans are part of a giant neural network which is used as a computer by aliens. It is in no way related to the institution, it is al about the freedom to believe whatever you want.

Now lets take a closer look at churches. Churches (in the broadest sense of the word, also including Mosques etc.) are in essence nothing more than a community of people who share a religion, right? Now, while these communities are important for the believers, they are in no way required to believe. I take it you agree, if not, please say so, because I believe this is completely logical. I could set up my own religion and practice it without ever having to go to a church. So, with this in mind, I state this: Taxation is in no way impeding the practice of religion. I can say this because of the distinction between the institution and the religion.

Now let us look at the first amendment. It states in relation to religion:

Congress shall make no law respecting an establishment of religion, or prohibiting the free exercise thereof

Let us look at that statement. It states that we cannot prohibit the free exercise of religion. OK, so I already showed that by assuming that there is a difference between religion and the institution. Now let us look at the first part. It states that we cannot make any law respecting an establishment of religion. So how about religions without churches? Like the Wiccans. Do they get tax exemption? I'd say probably not, because they don't have a church. Mind you, they do have communities, but they don't collectively own a building. People who believe "there is something, but don't know what"? No tax exemption, since they don't massively organize. We can see that in this isn't according to the first amendment, right? Either in your own words, if tax is indeed a form of control, then all these people would have to get tax exemption, otherwise it would not be according to the second part of the first amendment or when we take the route I just constructed, then the state "picks favourites" by not taxing the big religious institutions, which is also in violation of the first amendment.

I hope I have made my train of though clear. I am of course no legal expert, especially not of american law, since I am from Europe. Also, it is past midnight over here, so it might be that there are some errors in my post.