Sony HD hmmm - not really. The best tuners (see website FMtunerinfo.com) with most natural sound are vintage ones and come from the 70’s & 80s. Top of the line models from Kenwood, Sansui, Marantz are highly sought after for a reason. Look at the prices! However, you can go down their lines and buy great ones for a only a song.
The FM-5 was an ok sounding one out of the box (the tube FM-3 was better), but Frank Van Alstine had a wonderful mod for it many years ago that included replacing a lot of the disc caps with silver micas among other things. After he finished with mine, it sounded terrific. I still have it. It was way better than my Magnum Dynalab Etude that cost $975...25 years ago - I sold that. See if he would do a mod and how much it costs and compare to the ones on the site above.
To get decent sound on FM you need a decent antenna and an alignment on your tuner. The Terk will help, but get the antenna up as high as you possibly can if you want decent signal strength. Before you spend any money, make sure there are quality stations within a 40 mile range that you actually want to listen to. Your problem in LA will be multipath. Good stations can be few and far between but classical ones are generally better ( FMfool.com might help). Good luck.