Just to check back in...
The problem was none of the above.
I use a GE Smartwater filter system, and while the filter was new, the quick connect fittings are as old as the fridge (about 7 years). They're set up to automatically shut the water off if they're not connected to a filter. Apparently the check valve (or whatever it is) that accomplishes that purpose failed and was blocking flow into my filter. Fortunately I had another set of those fittings handy and had it fixed back up in 5 minutes.
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