The way I understand it, an amplifier with bluetooth capability means your phone is (wirelessly) connecting to the amp rather than the car's radio.

You say you don't want a phone payment, but you don't need the 'latest and greatest' phone either to have bluetooth capability. You can probably pick up a slightly older model for cheap. I'm currently using an i-phone 8 plus (i-phones are up to, what? 13 now?) and I stream music all the time with no issues. I bought my phone when it was new, but it currently sells for under $300 (just using my phone as an example). Also, if you don't mind a few commercials, you don't have to pay for the music streaming apps either. I use i-heart and spotify. (I've found spotify to be more reliable)

Or as mentioned also, you could just get a portable bluetooth speaker. You won't have surround sound obviously, but the depth and clarity of some of the speakers out there is incredible. One plus on going this route is you can use the speaker anywhere you want, not just the car. I have a cheap speaker I picked up a few years ago, and I use it at work, in my truck, in my shop, a lot of places.

Hope this helps a little.