Woeful AM on a radio is depressing and downright unacceptable. If a radio has lousy AM, no other features, no matter how admirable, can redeem it. Fortunately, there are several excellent AM radios out there. I will list the ones that not only pull in AM stations with remarkable power, but lock on the stations without drifting and have a minimum of background noise.
Number One: Kaito 2100. Perhaps the best general purpose radio available because of its unparalleled AM and FM. My only complaint is that you have to toggle through a preset button, only moving up, to go through your presets. A minor peeve: The buttons require firm pressure. But still this is the AM king.
Number Two: Sangean PR-D5. This features a huge 200mm internal AM ferrite antenna, and it shows. Outstanding AM. At around $70, I'd say you can't get a better radio for the money. I'll be curious to see if the new Sangean PR-D7, featuring a monoaural 3-inch speaker, unlike the dual 2.5 speakers on the PR-D5, is implanted with the same AM antenna.
Number Three: The now defunct Boston Acoustics Recepter. BA does an amazing job with their tuners. The sound quality and absence of background noise is amazing.
Number Four: Boston Acoustics Horizon Solo. The AM on this new radio may even be better than its older brother the Recepter. I've had a few minor problems though and readers have been reporting MAJOR BUGS. If you buy one, keep the receipt and make sure yours isn't full of gremlins. I suspect newer models will get better. For a high-fidelity bedside radio, the BAHS, in the absence of bugs, may be the best of its kind.
Number Five: Sangean WR-2. This is the clock radio I use. AM is outstanding. FM is better in my new model than it is in the one I bought back in 2005. Highest recommendation.
Number Six: Eton S350 DL. Outstanding AM and FM and very similar in look to the Kaito 2100. However, the 2100 does a better job of hiding the birdy noise when AM gets weak at night or in hard-to-get stations. Still, a great AM radio.
I know you had trouble with your Tivoli Model 1, but I still think anybody looking for a high-end radio without presets or clock owes it to themselves to at least try one... They do sound great when they manage to grab and hold the station you want (which is 100% of the time for me).
I never experienced the FM drift problem you reported -- in fact my impression is that the radio even improves after "warming up" a bit. The FM reception on the BA Solo I tried was not as good as on the TM1. I never tried the Sangeans because I can't find any in store.
My impression from reading too many reviews on-line is that all the "usual suspects" can be excellent radios. Reception may be highly dependent on your particular physical settings though.
I completely agree with Cyril. Don't judge a radio based on any reviews, even mine, until you've tried the radio yourself because your environment is the final determiner of the radio's performance. Case in point: One Amazon reviewer actually wrote that the Kaito 2100, the best reception-radio I've ever owned, gave him "lousy" FM. I have no doubt that the problem was his environment, not the radio. If 99% of Kaito 2100 owners rhapsodize about its great reception powers, then there's a high probability that it will perform to your expectations, but there are no guarantees.
So to be fair to Tivoli, you should give the very well-built Model One a chance before dismissing it. And we can apply this principle to most radios.