I don't know what you have for the AV budget, but I wouldn't worry about the main speakers producing bass- if you live close to others, a sub's level will need to be monitored, in order to prevent annoying them but if you want real bass in the garage, bookshelf speakers aren't where you'll find it unless they're placed where three panels meet (two walls and ceiling) and they're able to handle high excursion. If they can't, you'll be disappointed- the bass response will be lacking unless the woofers are large.
The flooring will do almost nothing to help the acoustics- the pattern is far too shallow for that.
I did a garage with two JBL C-28 speakers and a sub and it rocks. The output is limited to the receiver's power, but if you want a simple system, look at the Yamaha MusicCast WXC-50 and find a power amp that has a 12VDC trigger. I have used Dayton amplifiers from Parts Express and they're good- one is 9 years old and in a fitness facility with a Yoga studio- I bridged the channels in the Yoga studio because it has a pair of Polk outdoor speakers but the main room, which is 35' x 65'x 14', has eight 8" in-ceiling speakers from Proficient- she had a tight budget but they sound good. I also installed a JBL Control-Sub2210 dual ten inch sub on its own Behringer A500 power amp and put a Behringer FBQ-1502 equalizer with sub crossover. Since I have everything set to mono, one channel of the EQ is for the main room, the other is for the Yoga studio. She uses a wireless mic and that required a small mixer, but for your garage, the Yamaha WXC-50 can be used for all of your sources- I have one and am using it as my preamp, with an old Sony ES integrated amp connected for its phono section.
The Yamaha can be mixed to mono, it has filters, compression, BT, Airplay, ethernet port AND WiFi (I strongly recommend wiring ANYTHING that doesn't move) and it streams whatever you want, or uses the phone as the source. It can also use the USB port for outboard storage devices. The app works great and I have had mine for almost two years- I like the sound, flexibility and extra input/output jacks- these are the main reason I chose it over Sonos.
I also did the audio upgrade for a school gym-
Crown CDi1000 power amp, eight JBL C26D 70V in-ceiling speakers and the same JBL Control S2210 subwoofer- that system also needed a mixer, but the Crown amp can be connected to the MusicCast piece just as easily but since it's a commercial audio piece, it doesn't use RCA, 1/4" or XLR jacks- it uses Phoenix connectors (also called 'Euro-style), but that's no big deal. It can be used with standard 2-4-8 Ohm speakers, or 70V/100V speakers in a commercial/industrial installation. It also has a DSP, which handles stereo/mono, limiting, level controls, 5-band parametric EQ for the input/8-band parametric for the output; it has a pink noise source for each channel, the controls on the face can be locked out and full setup is done with a computer via USB cable and Audio Architect software (which is really cool!). The gym has a full basketball court + about 12' on each side, a stage with heavy curtains at the front and the ceiling has a shallow slope, from around 16'-24' at the center.
If you don't need 500W/channel, you can use the systems as a model and buy what you do need WRT power- the QSC amp in the link should be plenty- it's actually rated at 600W/ch, but it's less money and doesn't do quite as much-
https://www.zzounds.com/item--QSCGXD4?siid=167722&gclid=Cj0KCQiAkfriBRD1ARIsAASKsQKarNUikEjUzUfEqKSRXhTPVihUlBiq5vJh-h4nZYLiiPLjviJ3mhIaArB4EALw_wcB