Sometimes Tortoise requires me to do cleanup for no good reason - meaning I didn't abuse / kill it / stop it in the middle ... IMO this is "normal opertaional flow" and shouldn't require a manual cleanup.
Is there a way to not show these cleanup requests to the user but rather just do them silently under the hood and present a consistent experience to the user?
Actually it's not TortoiseSVN, but svn itself which requires this. It means that something did get messed up. What and how is hard to answer with the facts you've given.
The best might be you try to work more consciously so that you remember what you did immediately before this happens and watch for a pattern there. If you found one, maybe someone could answer your question.
I have seen this before. I believe there is a use case that involves renaming and moving entire directories that really confuses SVN. The last time this happened to me the svn status on one of the directories I moved started showing up as "nested" and when doing a commit SVN failed saying my WC was locked and suggested using the cleanup operation. Of course, that did not work and it was quite an ordeal trying to figure exactly what I needed to do to resolve the problem.
I have also seen this when you are using two different SVN clients and you commit with one and the other requires a cleanup before it will complete operations. This happens to me occasionally when I commit with TortoiseSVN and I have Visual Studio open at the same time. In some cases AnkhSVN seems to require a cleanup when I try to update or commit with it the next time. It does not happen often though.