Tail light does not operate from OEM alarm
The circuit for Tail (Running) Light goes through TWO fuses - Fuse # * & Fuse #10 - if you had no power coming from the Key-Switch then is almost certainly Fuse #8 is bad
The Origin for Turn Signals from the Alarm is different vs the Bar Control Switch - the fact that they flash with alarm proves the lamp fixtures/bulbs are working so that is positive information - but the circuit to flash from switch is different
The power for the Turn Signals originates from Fuse as the Brake Light & Horn - Fuse #4 - so first question there is do those other devices work? If none of them work, problem is Fuse 4
If brake and horn both work, then the most likely issue is the turn signal relay - did you replace the original (was IT an OEM?) with another original? Or with an after-market one? If latter, what is the part number of the relay you have installed?