While I appreciate the response from A. I. Breveleri, it still left me wondering why exactly the specific behavior I described was happening. I therefore kept digging on the internets until I finally got past all the how-to-fix-it advice to a plausible explanation of how-it-works (and why it doesn't when it doesn't).
The key misconceptions I had were [a] when conductors are heated their resistance to electric current flow increases (this is true for cables and transmission lines) and [b] there are electronic parts involved in opening the gas valve which means that it's either open or not.
First, the ignitor. It typically (but not exclusively) uses silicon carbide [SiC] as a heating element (https://en.wikipedia.org/wiki/Silicon_carbide). This is an exteremly hard and durable ceramic-type material (think harder than exterior grade porcelain tile) about which https://matmatch.com/learn/material/faq-silicon-carbide says "Silicon carbide, in its pure form, behaves as an electrical insulator. However, with the controlled addition of impurities or doping agents, and because SiC has the necessary resistivity, it can express semi-conduction properties [...] electrical ageing, which is the increase in electrical resistivity, could very well affect silicon carbide's electrical conductivity properties. Ageing is said to be related to the oxidation of the material."
So the way the ignitor works is as a variable resistor in series with the main power, thermostat and gas valve. As it heats it simultaneously glows brighter and reduces its resistance to the flow of electric current, thereby pulling that increased current through the gas valve assembly. (https://en.wikipedia.org/wiki/Electrical_resistivity_and_conductivity#Resistivity_and_conductivity_of_various_materials)
Once that flow enters the 3.2 to 3.6 Amp range it is supposed to activate the gas valve. What can go wrong with this arrangement is that, through oxidation ("rusting") or depletion of the additives used to make the SiC conductive (not sure), the conductivity is no nonger as good as it once was and the element no longer draws the required 3.2 Amps. It is not "weakening", it is returning to its "insulator" nature and this also explains why the decay is typically gradual.
Now the gas valve (http://gofarservicesllc.com/xyz_do_it_yourself_guide/oven_stove_range_cooktop_chapter_6 and https://www.appliancerepair.net/gas-oven-repair - neither is likely the original source).
As described in the linked pages, it is a purely mechanical device (electrically assisted but no solenoid/circuit board/etc) in which movement is provided by a bimetal strip. (For anyone who is not familiar with the concept, this is simply a strip of two dissimilar metals permanently bonded/sandwiched together. The metals are selected for their response to heat; both will expand but the rate of expansion of one in response to a given change in temperature is higher than the other, causing the strip to curl away from the side with the greater expansion - they were commonly used in old-style home thermostats with a mercury switch for instance).
To get the strip to curl enough to open the valve, it is heated by a low-resistance coil through which the ignitor ac current is flowing. The amount of heat produced by the coil is dependent on the current flow and it is rigged so that the valve will open at the set 3.2 Amp flow rate. This is such a rugged and low-pressure assembly that it is hard to get a mechanical failure but the coil can burn out. This is why the only thing to really check is that you have continuity through the terminals - if you do it's probably fine.
So now to my situation. The most accurate way to diagnose this is to use a (multi)meter with the capacity to measure in the 10 Amp (ac) range. Remove a clip connector from the terminal on the safety valve closest to you, connect the meter in series between the connector and the terminal and turn on the oven. The current should quickly rise to and remain in the 3.2-3.6 Amp range but in my case it rose slowly and stalled between 3.02/3.03, stubbornly refusing to go higher. Also, as it crossed 2.99 Amps, the gas valve opened and the burner lit but went out when the current stalled.
I still can't quite explain the initial ignition (after repairs I realized it was not actually releasing a full gas flow) but I assume it's some sort of spring/inertia/hysteresis effect where the increase/change in current flow opens the valve but the failure to maintain the increase trips it off(?). Tapping/blocking the jet apparently was just enough to settle the valve at a new equilibrium. Either way, the subsequent partially open behavior is what I would expect for something that is so close to normal operating conditions, just not hot enough to move the strip that last little bit to unblock the valve fully.
Replacement ignitor heated quickly and settled peacefully at 3.48 Amps for a happy ending.