For myself, and the students I taught, I reserve a Vx climb for those rare situations in which it's really necessary to clear an obstacle--and realistically, that happens really, really rarely. Sure, you impress the troops with your incredible steep climb, but what else do you do?
Well, for one, you eliminate the possibility of landing on the remaining runway (unless, of course, it's a pretty short runway to begin with).
For another, you shorten the necessary response time before the airplane stalls (is that 3/4 second, 2 seconds, 3 seconds? Everyone is different in their response to a sudden emergency).
I much prefer a climb out closer to Vy. OK, I won't likely have enough altitude to turn around if the engine quits off a 5000' runway, but I don't expect to. How much altitude do I actually need? Not counting my reaction time, I have tested it over and over in my airplane, and I need a minimum of 670' in my airplane with only me aboard to complete a turn-around which includes an S turn that brings me back to the runway, using a 45 degree bank. If I'm climbing out at somewhere around 85 mph and 500 fpm in a light headwind condition, if I'm climbing straight out, I'll be about 2500' past the end of the runway at that minimum altitude. But I need to be higher than that and farther out, because I will likely take several seconds to react. So my personal rule is that unless I'm at 1000' AGL when the engine quits, I'm not going to try it.
So what are the alternatives? It doesn't have to be straight ahead. At GXY, I usually take off on 10. Early enough, there's a turn to 35. After that, there's a newly constructed nice wide straight highway. Both require a sharp turn, to the left for 35, either way for the highway. There's also the old road that essentially parallels 10/28, although it's kind of hilly. And then there are fields. Any place within the airport environment is likely to be relatively flat, and that's true of many airports. So it's not necessary to make a complete turn around, or to necessarily go "straight out" to make a safe landing.
Unfortunately, the natural tendency when the shtf and the fan stops is to yank back to keep the airplane in the air, just the opposite of what is necessary. That's the same natural reaction when an airplane is rolled inverted by external forces, such as by wake turbulence of another airplane. So it takes a concerted effort to push the yoke/stick to keep the airplane flying, to overcome that natural tendency. That concerted effort comes from practice.
I think every pilot should practice the "improbable turn" to find out what it takes to actually accomplish it--but at a safe altitude, using an artificial hard deck as the virtual runway. For anyone who hasn't tried it, it will be a revelation--it sure was for me.
Cary