1" at 5 yards translates to 4" at 20 so whats the problem. Neither IDPA or USPSA require you to hit quarters. Both games use approx 18" x 24" silhouette targets with 20 yards being a long shot. If you miss the occasional head shot, so what. Head shots are usually at closer ranges.
I at 68 yrs. have 20x200+ uncorrected vision and have to work with blurry gun or blurry target but still manage to stay out of bottom place in our local USPSA shoots. My accuracy is suprisingly pretty good but speed suffers from excess weight, athritis and focusing time. It doesn't stop me from having fun and learning how to handle the guns a bit better. Last time out I ended up mid pack within all combined classes using a stock M&P40 with IWB holster.
Find and join an IDPA or USPSA/IPSC group and enjoy a fun sport while learning a thing or two on the way.