Various comments already pointed to problems that could come with a rough error-logging implementation. Because he used AJAX for every error, it creates a (unnecessary) strain on the network/server when in production. Besides this, the usefulness of the data logged may vary heavily. Therefore I suggest you do not log everything, like chrisacky points out. What exactly to log is an interesting resulting question.

I do fancy the idea of logging problems client-side, because they result in users leaving your site but leaving you without a clue what happened.

I think chrisacky has a good solution there as well. You just start to filter things you know you can't do anything about.

The approach to rolling this out then might be to do it for only a small % of users at first so that it isn't overwhelming. You include more users as you notice that your manually created filters work.