You have to evaluate what the problem is. If the risk posed by the high-speed abort and subsequent off-roading (if above refusal speed) is less than taking the emergency airborne, then you have to go with the option of what poses less risk to you, any passengers/crew, and the aircraft.
This is why we are professional pilots and not robots.
It really depends on the aircraft, that aircraft's systems/capabilities, and what the possible failure modes are. For example, in the F-15E where I have antiskid brakes, a tailhook, and at least two arrestor cables on the runway, it is possible to abort above "Max Abort" speed (which is this jet's term for refusal speed) and not end up in the dirt off the end of the runway. For certain catastrophic failures, this may be a less-risky option than taking a sick jet airborne and dealing with a really, really severe problem. In some cases it's even a better option than just getting airborne and ejecting from a really sick jet.
So, V1 is NOT necessarily a binary decisionmaking speed in all cases.
I'm not really sure that you can compare the civilian vs. military procedures for this kinda stuff, as you guys have a lot of other factors going for you. As you said, in the end, you can always give the plane back the tax payers. In fact don't you have a zero zero ejection seat on that thing, allowing you to punch out while the aircraft is rolling down the runway?
While I agree we're not robots, the standardization is there for a reason. In many ways, in the last 50 years especially, we've found that we need protection from ourselves. Simply put things can be happening too fast to consider all the factors we're discussing in this thread while you're chewing up runway at an incredibly high rate of speed, so with that we have briefs. The standard brief I'd hear from captains usually included this:
-Below 80 knots we'll abort for anything.
-Above 80 knots we'll abort for engine fire, failure, loss of directional control or a control surface failure.
-At V1 we're taking it flying unless the airplane won't, at which point we'll take what we've got on the abort.
But the fact of the matter remains, you need to accept that you've got a good chance of going off the runway after V1. Now as you said, that might be preferable to taking the plane airborne, but what if you're on a 7,000' runway with a 100' drop off at the end of the runway? I'd take almost any problem into the air as opposed to taking that drop off at the end of the runway, as that's going to be certain to cause a hell of a lot of damage to the aircraft and I'm willing to bet get everybody on the airplane killed.
So let's take an example I think folks were talking about; The CHQ EMB-145 incident at JFK. These guys had a problem with their elevator (gust lock didn't release properly, but checked ok in the cockpit to the points. They've got new procedures to check it now), couldn't rotate and I think ended up aborting at something like V2+15. Now by the time you hit V1, you're accelerating INCREDIBLY quickly, so I'm willing to bet they were probably V2+15 within 5 seconds after V1 (it's just a guess after flying the aircraft for a year). In those 5 seconds you'll end up thinking:
-V1, rotate....rotate.....rotate, why isn't the other guy rotating.
All the while the flying pilot is thinking.
-V1, rotate....rotate....rotate....what the ####.....WHAT THE #### IS HAPPENING!?!?!? ABORT!!!! ABORT THE ####ING TAKEOFF!
I mean honestly something like that is so out of the ordinary that it's incredibly hard to train for, and if it takes you only 5 seconds to realize that the airplane isn't going to come off the ground, on a lot of runways you're already off the end of the runway and probably dead.
The point being, fly what you brief. The airplane doesn't fly at V1? I'd say that at that point, you need to be a robot. You briefed that if it won't fly at V1 due to a control failure, then you're going to abort the takeoff. Don't try to over think the situation and get yourself killed, get the airplane stopped as soon as possible. Those guys in JFK got lucky they weren't in AVL when that happened, and they were lucky they had so much runway to play with. They would have had fractions of a second to make the decision to abort AT V1 if the airplane decided to not fly, and if they were dicking around thinking about all the possibilities of what could be happening and whether it's a prudent decision to abort, they'd be dead.
I mean to be honest, there are very few situations in an airplane when something goes wrong and you need to be on the ball 2 seconds ago or you and everybody in the plane is going to die, but things going wrong on the takeoff roll can produce situations that will require those kinds of reactions. Most of the time, when something goes wrong, I stick the time tested policy of winding the clock. Unfortunately, with how quickly things happen in a transport category aircraft on takeoff, with a limited number of options on how to get the airplane into a safe situation in an incredibly limited amount of time, we need to be robots then, or at least I feel we do.