I use the slow-motion function of my iphone's camera. Is it a reliable method?
For lower frequency flickering it should be. I can't see any plausible reason why it would fail. Even if the source flickers at the same frequency as your phone is recording at, you will still see something because of the rolling shutter effect.
instead the LED bulbs even if at 2200K and with filaments (vintage style) give me problems
Might be persistence: halogens take time to go from bright to dim, whereas LEDs can turn on and off in an instant. I don't know about your TV, maybe there are some capacitors that give it some persistence.
Could be that your LED bulbs flicker at high frequency, and that is a specific trigger for you.