I have to agree with some of the above, your budget is a bit lopsided towards the sub. And if you were going to get the Monolith sub, it should be the ported version. It is a great sub though.
If you want to maximize sound quality on that budget, I am going to suggest a very different route. This route will get you a system that sounds way better than anything you can get that includes an AVR. I think it would blow the doors off of any of the above-recommended system, but it is a bit unorthodox.
First, depending on your source, I would get an HDMI audio extractor.
Here is an inexpensive one.
Here is one that is a bit pricier. Connect that with the optical out to a good DAC with a TOSLINK optical input and Balanced outputs.
Here is a very good and inexpensive one. Connect those to some good powered monitors using XLR to XLR cables, and my top choices are the Kali LP-8, PreSonus Eris E8 XT, ADAM Audio T8V, or the JBL LSR308 mk2. On the Schiit Modius, use the unbalanced RCA outputs to connect to the sub. The Monoprice THX 10" Select is a great sub, but the
Hsu VTF-2 mk5 offers quite a bit more performance while being only $100 more.
This cost of all that would be about 1200 to 1500 depending on what components you select exactly, so it can be about the same price as you are already intending to spend. The disadvantage is that you are stuck with a two channel system. The advantage is it is just going to sound way better and be far more accurate than any surround system with a receiver that you can buy for the same price. It will also have far more dynamic range, like you can get near true THX Reference levels if you select the right stuff.
I have experience with many of the products mentioned so far in this thread and understand the performance class and product class very well. There is no way I would trade the system I outlined above for anything else mentioned in this thread- and I even like most of the products that have been recommended so far. But it's just not on the same level.
Something else to consider is that the main advantage of a surround sound system isn't the surround channels, it's the inclusion of a center channel. The center speaker anchors dialogue to the center of the soundstage for those who are listening at odd angles, i.e., they are off to the side. The problem is, all of the center speakers you are looking at are enormously compromised; they have real response problems off-axis which pretty much defeats the point of a center speaker to begin with! Just skip the center speaker! You are already on a tight budget, don't spend money on a component that has middling returns.
As for surround channels, contrary to popular belief, surrounds don't really add much to a movie-watching experience. They are a distraction when they are not dialed in correctly (which is like half the time), and they only add subtle immersion effects when they are correctly used. They are very disposable, and if someone just disconnected them while you were watching a movie, you would never even know they are gone.
To boil the above two points down to a sentence, a good two-channel system is way better than a mediocre five-channel system. Trust me, if you compared the two-channel system you can get in your budget with the five-channel system you can afford, the difference would be pretty stark; you would go for the two-channel system every time. A powerful, full, accurate sound with true deep bass is going to be much better than a constrained sound with noises coming from the sides of the room.