coaxial cable is the full video & audio signal which is processed through the TV. The TV typically has a 5 watt or so audio amplifier. That is not much, just enough for typical built in TV speakers. Even if you know how to tap into that, the 5 watts or so audio is not enough for a couple of external speakers with decent volume.

If you did know how to tap into the TV audio, I guess self powered speakers could work. You would still not have an entirely satisfactory volume control & might have to always adjust the external speakers volume at the speakers themselves.

The typical solution is to use an independent power source, usually an A/V receiver. The power amp section provides the power. Its volume control performs the pre-amplifier control functions such as volume.

Your amp/power source needs actually would be fairly modest. A $225 receiver would provide all you need. The higher priority is actually the speakers. No speakers are perfect yet each brand tends to have its own sound character. The electronics are designed to operate while remaining neutral, that is not introducing sound change & by and large, they do so. Most all the difference comes from the speakers.
That is why getting out to audition them is so important. Otherwise its like buying a pig in a poke.