Haven't we all experienced this at one time or another.
I had the pleasure of living in a one minyan town, to which there are many beautiful advantages. However, there was an unfortunate streak of deaths spread out 10 months apart that led to the morning tefilla consistently being led by the most tone-deaf, Hebrew illiterate mumblers for a good few years. It was painful - dentist office painful. I was able to find an antidote from quitting the minyan from the Ramban's famous letter to his son:
Therefore, I will now explain to you how to always behave humbly. Speak gently at all times, with your head bowed, your eyes looking down to the ground and your heart focusing on Hashem. Don't look at the face of the person to whom you are speaking. Consider everyone as greater than yourself. If he is wise or rich, you should give him respect. If he is poor and you are richer -- or wiser -- than he, consider yourself to be more guilty than he, and that he is more worthy than you, since when he sins it is through error, while yours is deliberate and you should know better!I found a way to see these daveners as making a significant contribution to the tefilla, and humbled myself before their off key nusach. Right before Kol Nidre on Yom Kippur we recite a permission, a public calling out of a policy to daven with sinners. "Al Daat HaMakom" may have entered the traditional mahzor after the Spanish inquisition and the prevalence of crypto-Jews. However I think it is refreshingly honest of us to recognize that although we are striving for purity and perfection on the holiest of days, we are also among wanton sinners, annoying people, and stiff-necked ideologies. How to pray with annoying people? It's easier said then done. I guess the first step is to ask yourself if you are that person and be the change you want to daven next to in this world.