Then I bought two power banks (2A and 2.1A) to supply power to my RPi.
I specifically bought one of these because it supposedly contains four of these, which are supposedly used in robotics and recommended by the manufacturer for, e.g., (presumably small) electric vehicles.
That was probably pretty ignorant of me electrically speaking -- I am now thinking what's important is how the voltage is regulated, not what the power source is on the other side. Power banks are sold fairly explicitly for charging, a use case which is going to have greater tolerance for fluctuation than actually operating an electronic device, and what's more, probably often (as in the Gogyre G2 example) include "over current" protection which may curb sudden instantaneous draws of whatever magnitude, and that's exactly what a CPU revving up and down will do.
Anyway, that's speculation from an admittedly (electrically speaking) ignorant person, but what happened with that battery and the Pi 3 is the undervoltage indicator (a little rainbow square upper left if you are using a display, and a flickering power led) does intermittently indicate an issue. It does not stay on for more than a second or so and there does not seem to be any real problem, but I haven't used it that way beyond a few casual hours.
Does that mean it's unsafe to run it like that?
Probably not in the sense of causing physical damage to anything, as the voltage is internally regulated and it is over and not under that is more likely to burn things. Although I think current surges can be side effects of sudden voltage drops, the Pi will presumably shut down if things fluctuate that much.
Note that if the system does shut down arbitrarily this way, when the battery has plenty of juice left but is overwhelmed during sudden bursts of activity, if the SD card is part of that activity, you may end up with filesystem corruption. But that is not permanent physical damage, it is just a hassle.
So I intend to do it when I feel like it anyway and accept that risk, but I'm also willing to accept what I consider to be a much more remote risk of frying a pi (and remember, I said I was ignorant).
To summarize: Don't expect to power the Pi 3 with a power bank and not have this happen.