Yes, it would be dangerous.
The power supply embedded inside your Honeywell controller is not designed or documented to provide power to arbitrary external devices, and the exposed terminals are meant only for the purpose documented.
The schematics in the manual are intended to provide a better understanding for the purpose of installing it in the intended way. They are just enough for that purpose. They are not intended to provide the basis for reverse-engineering, such as for tapping power off the embedded transformer.
Technically it might work: for an engineer tinkering with this controller for fun in a workshop, it will probably work.
Practically, in the sense of leaving this unattended in a home for months and years hoping nothing ever goes wrong it is untested, unproven, undocumented and therefore: dangerous.
Danger is not binary. This is not a case where you need to choose the least dangerous path forward. You have innocuous alternative approaches:
- Buy a battery-powered thermostat. True, that rules out ones with color displays and Wifi.
- Install a new transformer near the furnace to supply thermostat power
- Install a power supply (wall wart) or a large battery pack near the thermostat to power it.
Many smart thermostats will take power from the separate A/C side and if you don't have A/C will allow you to connect a dumb power supply for the sole purpose of supplying power, via Rc/C.