I don't know about the 2017, but my memory of the 2013 schematic shows all the bulbs are driven directly by the BCM. The stalk just feeds commands to it. The clicking noise is created by the radio on commands found on the car's data bus. One way this becomes obvious is when using the 4-way flasher. With the car running, turn on the hazards. Now, turn off the car and open the door, killing the radio. What happens to the click? On my car, it disappears.
A 2005 is probably closer to a 1970's car in the way the lights are wired. If it's the kind of car where the brake lights do double-duty as a turn signal, then the stalk has to disconnect the brake line on that side to change the brake light to a turn signal. Lots of switch contacts. If that switch fails to connect, the lights don't work right and you can get hyperflashing.
That's my fear as well. But might as well try the cheap things first. I do seem to remember having a old bulb that didn't draw the right current and causing flashing problems. Bulbs are cheap, so it's worth a shot.