Well the circuit path to flash the turns from the KeyLess is actually quite different (does not go through the flasher or the turn switch) - however it does originate from the same fuse (fuse 4) - if fuse 4 were out, then the Key-Less would not work at all.
Question - Do ALL the turns flash from the keyless (ie ALL 4 corners) If one side does not flash it could point to an issue with one of the fixtures or bulbs.
Which gen of key-less is this? (Sorry I have so many out there now, hard to keep track of who has what version now)
I think you have the last model yes? Solid State version?
Can you tell if it at least initially comes on after you (effectively) turn on the ignition?
Test with turns, reset and test with brake - potentially you could have a problem with either of those circuits which is 'tripping' the solid-state switch (same as a fuse protecting essentially)
If ALL the turns are working (from the arm/disarm flash) then they would not be the cause; but if they initially work, then when you pull the brake they trip off, that would suggest problem with brake light circuit.
If it does not work at all after reset the ignition, then check the main key-switch connector.
The terminals of interest would be 6 & 8 - Orange and Orange/Green & Black/Blue on main side and white & orange on keyless side.
If you can access the horn terminal, (Black/Blue wire) - turn on ignition and see if that has 12V wrt ground (engine or batt negative - NOT the other horn terminal)
Then operate either Turns or Brake and see if the voltage trips off.
If you do not have 12V there at all (even before operating brake or turns) then problem would have to be at the key-switch connector - or the keyless switch itself.