"I could buy a DAB link to the FM for much less and listen to AM stations that way, but why should I?"
Because it's cheaper than the radio replacement/repair by far and you will get your preferred am stations back plus many more, and also with an improvement in broadcast quality over am which is akin to the improvement that am is over a pre-war cat's whisker set.
Once you've listen to the talk/sports output on DAB, you'll never again want to be bothered the constant retuning of the fading voice from a biscuit tin source that am is. It's a technology of last century, and belongs back there along with various other orphaned technologies that we've abandoned along the way, valve radios, compact cassettes, eight track, vinyl discs betamax, 35mm film, spectrum ZX. They were all ground breaking in their day... way back in last century!
You will need an external aerial though, the FM splitters, or the windscreen stick-on one are crap.