Decent tuner with a headphone output and volume control

I do the same thing as you in my bedroom system. I used to have a Bryston BHA-1 headphone amp with a SACD player and the Sony XDR-F1HD. I just recenlty sold the BHA-1. I get a great station on the FM dial, which sometimes plays vinyl too. The same station on the internet sounds like crap. I actually bought the Sony HD player solely to listen to the second HD broadcast from this station.

I bought a modified Sony tuner from this guy. I think I paid around $400.
https://www.xdrguy.com/xdr-f1hds-for-sale.html

HD radio is crap but the player also gets regular FM and the tuning is amazing. I also have a Magnum Dynalab MD 102 that cost close to $3K. The Sony is not as good but it is rather close to the same quality. For $400 it is a steal.

This unit does not have a headphone amp but what I will do, to replace the BHA-1, is get either the Headamp GS-X mini for $1799 or a second Benchmark HPA4 for $3K. So at the low end for around $2100 you could have a killer FM headphone system.

Not what you asked but likely better sound quality than what you are thinking.

Listening to my Meze Empy + MD 102 + Benchmark HPA4 as I write this. Awesome sound.

A little update. I now have both my tuners hooked up in my office with indoor antenna's on each and a Magnam Dynalab Signal Sleuth on the MD 102 tuner. 

https://www.xdrguy.com/xdr-f1hds-for-sale.html

The XDR on FM and even HD is great at picking up stations. At the moment the MD 102 is having problems clamping onto 1 station. I switched to the XDR and perfect reception on this station. This is an amazing tuner, especially with the mods I listed above.

A bit of posts on this tuner going back over the years.