2003 Mini mystery! Will start with assistance, but not staying running

[AtlasRS]

I'm helping a neighbor try to get his 2003 Mini back on the road. The car has been running rough for a long time but now refuses to start. Interestingly, it will start with a shot of starting fluid, after which it runs for 15 seconds or so, seemingly like normal. Then it dies. Trying to start the car with no starting fluid will result in one or two seconds of a running engine, tops.

Key details:
2003 Mini 1.6L, CVT, ~200k
Pressure in the fuel rails (confirmed via Schrader valve)
Clean MAF, throttle body
New spark plugs, new coil packs
Battery is ~5 years old but appears to hold a good charge.
New thermostat housing, thermostat, coolant sensor
Fuel pump and filter replaced <6mos

Prior to the no-start issue, the car ran out of fuel and would not restart. I found the original coil pack had no spark on one post, so we replaced the coil and the car ran normally after that, for one day. Mini died while on the road and won't start unassisted.

I'm so confused! We have fuel, spark, and compression, but it won't run! Unless we cheat with starting fluid... then it runs great!

PS—what's my prize for waiting 12 years to post?