You asked, "... when the blade uses a very acute angle to achieve sharpness... " Remember to separate in your mind sharpness and angle: they are two different things.
Keep in mind that the angle of the edge actually has nothing (or rather, very little) to do with the sharpness. A 20 degree inclusive angle (10 degrees on each side) can be a poor cutter if it isn't finished properly. And a 50 degree inclusive angle (25 degrees on each side) can be made shaving sharp. But for any angle, the more refined and polished the edge, the more durable the edge. So in that sense, it always makes sense to bring your edge to the maximum finish [for example, on ultra-ultra fine stones (2500+ grit) and/or a strop].
Otherwise, I'm not sure what you mean. If using your knife to chop, you will get cleaner and faster chops with a more polished edge. That edge will eventually get dull, yes, but an already dull knife will.... already be dull, so... it'll just get even duller.
As for how "fragile" an otherwise well-done edge is depends on the steel/HT. You can get much better edge performance and durability from a superbly heat-treated chef's knife in, for example, Hitachi Super Blue at 20 degrees than a cheap 420 Ikea chef knife at the same angle and refinement.
So remember to separate in your mind sharpness and angle: they are two different things. Remember that a more-refined edge is always better (although some people disagree with this statement). And remember that steel matters! Better steel will hold a more acute angle better.
Now then, what angle is too acute (or too obtuse) depends on your blade and what it is being used for. If you have a big, thick chopping knife (like the "camp knife" style that is popular among BF members), you obviously don't want to put too acute an angle on. A typical "user" angle is 40 degree inclusive (20 degrees on each side). And the more time you spend finishing and refining that edge (finishing on a strop, for example), the better durability and performance you will get from that knife.
Hope this helps