Gas clothes dryer not heating
tl;dr: running the dryer with the front face off causes the high-limit thermostat to cut off the heat

This may have been the ultimate rookie mistake, but at least my dryer is working now. I'm going to outline it all here in case it's useful to someone else.  The chronology of my DIY repair:


After disconnecting the vent out the back, I confirmed that the dryer was not producing any hot air.
After consulting debugging checklists like this, I disassembled the dryer to access all the components of the heating assembly. As described in the original question, I confirmed continuity and resistance as appropriate, and all individual components seemed to be working fine.
After talking with the fine folks at appliancepartspros.com (very helpful) they asked for more information about whether the igniter was actually lighting, and whether ignition was occurring. I did not know how to do this.  Apparently, many dryers have a little viewport near the burner assembly that will allow you to visually inspect things as the dryer is running.  My model did not have this viewport.  Therefore, I took off the front face/door of the dryer and reconnected all the components so I could do this visual inspection.
With the dryer face/door off, I observed what I wrote in the original description: "the igniter turns on, ignition occurs, and then after ~10 seconds, the flame goes out. After ~1 minute, the same thing repeats." 
As described in the original question, bypassing the high limit thermostat with a piece of copper wire fixed the "problem" -- the dryer heat stayed on as expected when tested with the dryer face/door removed.
Based on the previous observation in point #5, I replaced the high limit thermostat, thinking that it was tripping at a too-low temperature. However, that did not fix the observation described in point #4 above.
Grasping at straws, I also replaced the gas valve coils thinking they were somehow making the burner run too hot. However, that also did not fix the observation described in point #4.
I cleaned some lint residue off of the inside of the heater assembly. Again, did not change observation in point #4.
After almost giving up at this point, I went ahead and reassembled all components, including properly mounting the front face/door. That blocked my visual inspection of the burner assembly, but the dryer worked!


Here are my conclusions:


With the front face/door off, the heater apparently burns hotter (presumably due to more air flow), which causes the high limit thermostat to (appropriately) turn off the gas valve. That explains the observation in point #5 above.
That means the original heating problem may have been the same was what I described in point #4 above, but in all likelihood it was something different.
Something that I did in points #6, #7, and/or #8 nevertheless fixed the problem.  (I'm too lazy to put the original parts back in to isolate which part is was.)  But importantly, as described in point #2, the high limit thermostat had continuity at room temperature, and the gas coils showed the appropriate resistance across all pairs of prongs (as helpfully provided by the people at appliancepartspros.com). So unless the actual fix was #8 (which seems unlikely to me), one of the components tested fine but was still failing.


Again, hope someone benefits from this recap...