I believe the reason that running/tiredness doesn't work the way you've suggested is because that system is abusable, especially in shotgun games.
Basically, in a shotgun game, "Running" is equatable to "Combat mode". You use it just before a big fight to give yourself a huge advantage in dodging and movespeed with no drawbacks.
If your system was implemented, one could rip up a roomfull of demons and hang near the door, resting until tiredness goes away. If anything comes in to the room, they'll most likely be at melee range and'll get owned by your double shotty anyway.
And then, of course, you just repeat the process over and over whenever you enter a room with more than a couple of demons in it.