You didn't make clear which car is giving you the issue: the one with the e-flasher, or the one with the mechanical flasher?
If it's the one with the e-flasher, then it's probably not the flasher. They tend to be pretty reliable. Then too, it's probably not the stalk switch. Hard to believe that a flakey switch would behave that consistently; e.g., always come on for a second then go out with no blinking.
If it's the one with the mechanical flasher, then it's probably because the lights, sockets and wiring on the left have more/less resistance than the circuit on the right. Mechanical flashers are very sensitive to this, so replacing the m-flasher with an e-flasher will probably solve it. I doubt it's the stalk if it behaves consistently. But a diagnostic would be to switch the stalk to the left, wiggle the lever, and see if it behaves any differently. Or spray in some contact cleaner, and see if it behaves any differently. My guess is that it won't.