To me a significant skill use ought to have potential implications if you can for the most part just retry stuff let people take 20 and get it done is actually reasonable. And if you can just re-roll next round it better be very time sensitive or no way you want to spend that action surge why bother.
That's pretty much why it's never used out of an encounter. It might be an exploration encounter. Social encounters tend to lack that time crunch that an exploration might (typically doesn't) have. Combat is when time matters the most and that's why combat is when we see the ability.
If someone is dangling from a cliff and falls due to a critical failure (used in some campaigns) I would allow action surge to repeat the check before the fall, personally. It's almost like advantage on a check already so allowing such an application isn't a stretch either. I would allow that too if a fighter wanted to use action surge that way even if it's stretching the rules a bit.
Because when you use a high level slot you can have it permanently on.
That's not true. The ranger has to give it up to cast any other concentration spells, assuming the ranger even took the spell in the first place. All fighters have action surge. This topic is specific to battlemasters and all battlemasters have superiority dice and maneuvers.