Agile, in my understanding means "adaptive", and people often forget that part. If meetings or whatever else aren't working FOR YOUR TEAM, then don't do them. I think the retrospective is the only real cornerstone. It should happen, even if everyone just says "I think everything is working fine".
That being said, I think stand-ups create a "common time" where everyone can get be aware of what everyone else is doing, which removes that worry/distraction from everyone's mind at every other time. (I think being aware of what other people are doing is important, especially on a small company).
This is the best advice possible to people who have dysfunctional agile implementations. If what you're doing isn't working, stop doing it and try something else. The only thing that you should consider mandatory is a regularly scheduled retrospective. (And if people complain that the process keeps changing too quickly, you can always decide in the retrospective as a team to have less frequent retrospectives).
That being said, I think stand-ups create a "common time" where everyone can get be aware of what everyone else is doing, which removes that worry/distraction from everyone's mind at every other time. (I think being aware of what other people are doing is important, especially on a small company).