As the Wormtongue of decline, I would normally endorse something along these lines, but too many of my favorite game experiences have been the result of strings of good or bad luck. (The first, and most formative, being when a single archer defeated a full squad of archers when defending a fort in The Ancient Art of War. It was the first "emergent narrative" I encountered as a wee lad of 6, and it stayed with me. Obviously, the point-blank spray fire miss on a Chryssalid in X-Com, which then obliterated my squad through a series of mishaps, is another classic.)
Anyway, it is definitely true that people get sad when a series of bad rolls loses them a match. But I think you can mitigate this in various ways. The simplest would be to have bad-luck achievements, and have the highest of these achievements set relatively high thresholds ("Missed 100 shots with >80% likelihood of hitting." "Lost 20 battles in which you missed at least 3 shots with >80% likelihood of hitting."). Because there is significant overlap between "person who gets mad at missing and posts a negative review on Steam" and "achievement harvester," I suspect that if you calibrate these right, you will be able to eliminate a lot of the hard feelings. I realize this kind of stuff enrages the
Blakemoreland Hybrid Boss folks of the world, but I really think it's the best approach. (Just as I still believe that allowing players to autoresolve combat, a la Master of Orion 2 or Realms of Arkania, is a reasonable accommodation.)
Another, gamier solution would be to have outrageous misses accumulate some kind of "golden die" option (a la Frayed Knights) where you can get a critical hit if you've missed X number of times. This feels possible mood breaking, but I like it more than your proposed solution because rather than just smoothing things out, it makes the eccentricities of random chance fun in both directions.
Grampy_Bone's examples seem okay as well, but they may not really work. For instance, missing a Harried target when you have 90% chance to hit will just amplify the annoyance. Turning evasion into a shield generator (which is what I gather evasion pips basically would be) would significantly change combat, possibly for the worse.