FWIW: I use my 30+ year old SONY ICF 7600D SW receiver for just such a purpose.
Funny...I was just thinking of the Sony '7600 I used to keep by the bed and listen to the ham bands on at night...and that I'd hate to use it for QSOs! ;-)
Actually, I guess it would be doable, just not as convenient as a communications RX. Certainly the selectivity wouldn't be very good.
What boat anchors did you try, and what was the problem? I've got an old Drake R4 that I got for around 180 bucks that sounds great, and is a pretty good RX...works nicely with a Tuna Tin 2! I did have to replace the electrolytic caps in it, but that wasn't a bad job at all. It's ham-bands only, but SW bands can be covered with crystals to tune 500 kHz segments...or a "crystal eliminator" could be built or purchased. Maybe something like a Yaesu FRG-7 would work for you...
Also, there are some VFO/VXO type QRP transceiver kits out there, so not all are fixed in frequency.