It’s likely that the LEDs you are using do not draw the current of the incandescent bulbs (while still providing the equivalent luminance). It’s current that heats the element within the flasher that activates it; too little current, and it fails to operate.
If you wanted to go to the trouble, it would be possible to identify the resistance to shunt across the LED that would draw the equivalent current, while still operating the LED – you’d have to measure the current draw for both the LED and the incandescent and compute the parallel resistance value using Ohm’s Law.
Alternatively, you could get a solid state flasher – these aren’t thermally activated, they have a timing chip that turns current on and off.