Okay, for starters, I think we need to go over some terminology:
Amplifier: Take a line-level signal and turns it into speaker level, i.e. powers your speakers.
AVR: Audio Video reciever. Amplifier + switches between inputs, decodes surround formats like dolby and dts, crosses over your sub and main speakers at a particular frequency, and more.
Second, yes, you can get a home theater in a box for a little more money than you spent on your Ozaki. But, if I can create an analogy. Right now, you're driving a ford focus (Ozaki). It's a good car, gets you from point a to point b, but there a lot of road noise, the seat aren't very well cushioned, and it take a while to get to 65. You could trade in your ford, plus a little money, and step up to a Chevy Malibu, which is a slightly nicer car, but it's no Lincoln Towncar (and certainly no Lotus Elise). Not to mention that, if your not particular about your car, why'd you spend any money at all to upgrade to the Chevy? If you were happy with your Ford, then keep it. It still gets you from point A to point b right.
Before anyone gets cheesed that I made fun of their car, I drive a '91 Toyota Camry with 175k miles and collision damage. Chill out in your Ford Focus.
I fear that if you trade in for a $200 system, you're just going to be back in a month asking for advice on what you can get for $500. I'm not saying that the $200 5.1 system isn't any good, but you came here for a reason, instead of just sticking with your $50 system. There's something about your sound that you're not satisfied with, and I don't know if it can be solved for $200.
The systems for around $200 all have the same deficiencies. Their speakers are under engineered, they will not by as clear or dynamic as other, more expensive models. And, I don't know enough about your current system to be sure that it will power standard speakers at a level that will be loud enough to fill your room with sound without distorting.
The goal that you stated was to get rid of the bad quality sound. If that is your goal, I don't think simply adding new speakers will work.