Credit to thedragonborn456 for the idea/request.

Requires SKSE

This mod simply waits for the player to shout, and then it resets the shout cooldown. There's a half-second delay for technical limitations. It only works on the player. It triggers by shouting, not by changing the cooldowns, so it should be compatible with pretty much any mods or added shouts. It only resets the shout timer, it will not reset your ability to use the Dragonborn once-daily shouts.

No constantly-updating scripts, so installation/uninstallation couldn't be simpler and it's safe.