The approach you describe to measure speed from the motor position is essentially what subway cars use. Ignoring wheel slip, which is not a rare phenomenon, this approach works very well. However, when you defer maintenance, things operated in harsh environments _will_ eventually break down.
(Within the context of the issues mentioned by the linked 2018 NY Post article) Speedometers were (bizarrely) judged as non-critical parts (i.e., the car can still be used in service with it broken) because, after all, the signaling system will catch any over-speed, thus the repair, and more importantly the maintenance, of speedometers was not prioritized. Thankfully most of Cuomo's goons and bean counters have been pushed out.
As for the wayside speed enforcement, the author only briefly touched on the solutions to the problem described in the article, but it's known as Communications Based Train Control (CBTC)[1]. It's a moving block system (compared to current fixed block signals) that used train speed, track geometry, and the location of other trains to determine maximum safe operating speed.
I would argue that it's not "maddening" to control subway speed with electro-mechanical timing mechanisms, control lengths, etc. This was cutting edge in the 1920s & 1930s, and indeed some of the oldest signaling in the system is from that era (though thankfully, the amount is decreasing).
It is however maddening to decide in 1995, given other existing speed control solutions at the time (coded track circuits, CBTC, axle counters) to expand the use of these timers. But as the saying goes if you have a hammer, everything is a nail.
Even more maddening is how slow the subway's transition to CBTC has been. NYCT was an early leader, with the Canarsie line being one of the first brown-field re-signaling jobs (not to mention a 24/7 railway), and then the program just seemed to languish under management that didn't see CBTC's value or the need for modernization (could write pages on this). Thankfully the new cadre of people at 2 Broadway has put the CBTC program into high gear, with 4 (5?) lines under various stages resignaling at the movement.
As a bonus tidbit: the wheel slip issue mentioned above is fixed in CBTC operations with the inclusion of a free axle, equipped with no motor or breaks, thus never experiencing a lack of adhesion. Passive RFID balise's placed at known intervals (i.e. loaded into the train) allow the train to then audit (while in operation) how far its estimated position and speed have deviated from where it truly is. Some CBTC systems also have car-brone backups based on accelerometers or rail-facing doppler radars.
[1] https://en.wikipedia.org/wiki/Communications-based_train_con...