I have to agree, I may be a little misguided here, but if a radio has any sort of reasonable sensitivity, then the more important thing is the selectivity, adjustable bandwidths, and resistance to overload, less intermod distortion etc...I've had good results with the Tecsun PL-880 and a small loop nearby. The 880 can be found right at or below $100 used nowadays.
It's nice because there are so many audio bandwidth choices so you can really fine-tune the sound.
Another reason why I agree with the poster above- if they have a laptop, an SDR is a very attractive option for AM DX. user selectable bandwidth really helps with copyability. I have an HF+ Discovery and it's brilliant for AM but it's more expensive than your budget.
Sensitivity can be 'effectively' improved with a larger or better antenna.
Unless the radio will be used absolutely 'barefoot'. Which may be the objective here.
Thanks
Joel