What I was trying to say is that if a person speaks a language correctly, but has a poor accent, communication still occurs. It doesn't bother me that much when I hear heavily accented English, but I know for others it can be a problem. "Distracting" is not the same as "preventing". The poor accent is mostly annoying, and most of us would rather not be annoyed. But bottom line, if it's understandable, then communication happens, and that's what is important. :0)
The contradiction remains. You claimed that accent is
unimportant in communication. This is clearly bunkum. Accent is very important in communication in many situations. I've seen Welsh and Scottish programs on TV in which the dialog was in English but English subtitles were provided. I've also watched a film in which it was easier for me to follow the actors when they were speaking Punjabi, a language I know little of, than when they were speaking English, my native language, because their Glaswegian accents made the English almost impenetrable. All of this invalidates your assertion that accent is "unimportant in communication".