What colour ESP-01 are you using?
If the pcb is blue it will only be an older 512Kb device, where-as the more recent black pcb's are 1Mb.
Using the current V3 flasher on a 1Mb (black) device should get all V3 features.
But using V3 on a 512K (blue) ESP-01 will only load a stripped-out version capable of running in the limited memory... I don't know exactly what is omitted, but it might explain your problems.
If you are not wanting to use anything V3 specific, an option would be to flash your blue 512K device with the last V2 (Alpha 24) which runs fine with all of it's features in 512K memory.