Salt Lake is a great place to live for most of the criteria you mentioned. If you lived in Sugarhouse, the Avenues or near Liberty Park (I know a couple SP members that live in these areas), you're less than 20 minutes to the trailhead for great trad climbing in Little Cottonwood and Big Cottonwood (former is quartz monzonite i.e. Yosemite-style granite, latter is juggy quartzite a la the Tetons). Certainly would have access to asian markets, quick access to the airport (less than 20 minutes also), plenty of trails to run very close by (including ones that could be accessed directly from your neighborhood if you choose semi-wisely).
The big downside to Salt Lake is the lack of genuine alpine climbing, in the sense that I think you mean. There are some "alpine" climbs in the Wasatch which I have loved doing, but there are not a lot. Maybe 5-8 established alpine rock routes that end on a summit that get done very often and are not total choss. 3-4 of them are on the same mountain, with quick access to SLC though (Olympus). I might be underestimating, but not by much. Lots of spring/summer snow climbing and waterfall ice though.
The Uintas are great for backpacking and would be great for trail running and peak bagging, but have no alpine climbing.
The Tetons, Wind River Range and parts of Colorado are a 4.5 to 5 hour drive away (Sneffels Range in CO). These give you great alpine climbing options. The Tetons are enough to keep you busy for a decade plus, even if you're a quick, frequent climber as you seem to be. BUT, this is no closer than you currently are to alpine climbing, so it's debatable whether this is really an improvement.
That's my two cents as someone that lived near Salt Lake for 6+ years and is dreading the move to the East Coast. It's really great in almost every aspect of what you mentioned, but just be aware that for genuine Sierra-quality alpine climbing, you will mostly be traveling to the Tetons/Winds/elsewhere. Of course, it's central enough that you have access to a lot of ranges (8-9 hours to RMNP in CO, 8-9 hours to the Sierra, 4-5 to the Tetons, Winds, western ranges of CO, 3 hours to canyon country, etc). Anyway....