As an alternative to the standard crit mechanic, how bad an idea would it be to give the player an additional action after rolling a crit? It's kind of like a stunting mechanic (a la AGE system) but I'm not sure it would work in 5e.
As I understand this question it reads: Is getting action surge as a crit reward reasonable?
Considering classes that get extra attack, this could mean a whole extra round of attacks. Similarly, monsters that get multiattack could get another round of attacks off. Theoretically, this could result in a chain of crits and attacks.
This seems far more reasonable (although still strange) if you consider it being like the Haste extra action, where it is limited.
That action can be used only to take the Attack (one weapon attack only), Dash, Disengage, Hide, or Use an Object action.
A standard action can have a huge variety of uses. Hit dude & pick a lock, Hit dude and run extra far, etc etc. It makes more sense to keep combat rewards in their context.
Second point: Thematically, a crit is a very clear idea. You managed to hit something in a vulnerable location and you deal extra damage because of it. Explaining a crit as "I was able to run extra far that turn because I hit a thing really hard." is harder to pass off.
In my opinion, I'd leave crits as they are: landing an extra effective blow. However, you can definitely describe the extra damage any way you want. Nothing prevents you from saying the damage coming from a second flavor attack. A monk crits? While the creature was reeling you manage to land a followup blow.