If the goal is to provide the purest form of sport, then you choose the best 3 from the trials and go with it. No exceptions. Pressure is part of sport, and if you false start or no height in a jump, then that means you were not the best that day. Who's to say it won't happen at the Olympics too? This is my preference.
If the goal is to provide the highest probability of winning a medal for USA, then I think the best formula is something like a top 2 + 1, where the top 2 in the trials event get a spot no matter what. The 3rd is a flex spot chosen by a committee from athletes that competed in the event final at the trials.
The default choice would be the 3rd place finisher, but it allows for exceptional events such as a false start, a no height, taken out by a spectator, dog runs onto the track, etc. Not a fan of this though because I could think of scenarios where it's easy to game the system. For example, there are 4 sprinters within a very narrow margin in the pre-lims. The top sprinter feels like the committee would definitely pick him for the 3rd spot, so he false-starts on purpose rather than taking a chance of getting beat by the other 3 guys.