Be aware that ADAT is linear on magnetic tape, so you can't quickly jump to a specific spot. I've seen solid state backing track systems used on cruise ship theaters, but I researched them and really it was just quality hardware running an embedded version of Linux.
Maybe this ventures a bit off topic, but may I ask you about the non-computer requirement? Ableton Live is rock solid. For about 4 years now, I've used it weekly on a Mac mini (with the internal drive even!) to play backing tracks that span anywhere from 5 to 15 separate stems per song. Those are multitracked out to the console and MIDI triggered with an 8-button footswitch. In all that time, I've only had issues once and it turned out to be a corrupt set of stems.
The Mac mini is dedicated to the task, with all screensavers/backups/scanning/indexing/notifications/etc disabled, as is the thing to do on any production machine for live music.
I also sometimes play raw tracks from Logic and Reason on a 2013 Retina MacBook Pro. That's been reliable. Except for the very-synth-heavy songs that I haven't printed to audio yet - those can get a little hairy on the CPU. =)