an inverter probably draws 10-20 watts switched on with nothing plugged into it. a small CD/FM stereo likely us also 10-20 watts, so you're wasting about 1/2 the power just running the inverter. Now, if you have enough batteries, thats no big deal
me, I use a USB chargeable bluetooth speaker and my USB charged phone to play music, with Spotify. Most of the country, FM radio is a wasteland of garbage, and even tho I have 1000s of CDs, I've not played any in some years, since everything I want to listen to is on Spotify, and I can download it on my phone to listen to when I'm not online.