My math might be utterly wrong, but it looks like a Teensy in sleep mode consumes 0.23 milliamps at 3.3 volts, which comes out to 66 joules per day. The OP mentions that the mechanism provides about 13 millijoules per day, which is 5,000 times lower.
Teensy is nowhere close to most power efficient options out there. A power efficient arm can be running on micro or even nanoamps in sleep mode. It will of course depend on the rest of the circuit surrounding MCU.
you would probably get a lot more energy in cumulative milliwatt-hours per week/month from having a tiny 2 x 1cm size photovoltaic cell such as from a cheap desk calculator exposed to ambient room light, and something like that has no moving parts.
if we are talking about actual practical IoT things a single "big" solar cell (125x125 or 156x156mm) size is quite cheap, and you'd run the output of the PV cell directly into a tiny battery charge controller circuit configured to keep one 18650 or 21700 size battery at its float voltage.
I guess that tire pressure sensors use batteries. Always seems like they should just be able to harvest power when you're driving, and have plenty of juice to get past the idle times.