No, these issues are very much related. It is the very nature of proprietary software that you cannot inspect and modify it, so you cannot know if it will track you or not, and cannot fix things if you are.
(Inspecting outgoing traffic is helpful, but unless you monitor all activity all the time, and make the effort to actually understand every single bit that is transmitted, you can't be certain.)
Furthermore, some of these browsers explicitly do track you. For example, Internet Explorer, Chrome and Safari provide ways to sync your bookmarks, all of which track you - and some of them encourage you to do this, for example if you do not log in to Chrome it says "you're missing out". (Firefox also has a sync service, but it is encrypted on the client, so the server cannot read the information, and you can't be tracked.)