I cannot speak for other companies, but at NetJets, we have 91, 91k and 135 flights. Now for the purpose operational regulations, we fly under those separate flight rules depending on the type of flight (91 repo, 91k owner flight, 135 card holder flight). For the purpose of duty, you cannot start 91 and not count duty or anything like that. But we have different visibility and ceiling requirements for takeoff and landing under 91, 91k and 135.
As far as personal minimums go, there have been times when the FAA throws a reckless and dangerous at you, even though you weren't breaking a reg. If you takeoff without the means to land, is it legal? Sure. Is it safe? No. Is it potentially dangerous? Yes. That being said, in a two engine jet I am hardly worried about it. In a PC-12 you bet your ass I am worried about it, so the situation differs.
At the end of the day, is it really worth departing in a potentially dangerous situation that can get yourself killed? When I would face scenarios like that and my previous companies, I used those instances to fuel my effort to GTFO. I have had several times when a captain said "see, the tire didn't blow, I guess you were wrong, I guess it wasn't too bald." And I just scratch my head and think to two very specific instances where people died because of a blown tire. The odds of something happening are low, but they aren't zero, and 999 times thinks go perfect, but a professional pilot, and a professional company should want to plan for that 1 time where is something happens, you don't make the 5 o'clock news. Bending rules, busting minimums, not planning a safety margin has absolutely been the cause of hundreds of deaths in aviation.