I agree with you to the extent that mass migration can indeed cause problems for both countries involved.
However, I think you're missing the other - very important - rationale for trying to keep immigration controls as light as practical: That people should, in principle, be free to determine their own lives, including where they wish to live. Remember, ultimately, all immigration controls amount to Governments basically telling people where they are and where they aren't allowed to live. Often that includes by implication - preventing people from seeing their families, their friends, even their husbands or wives (And yes, that really does happen *a lot* in the UK today). Too many people blindly call for harsher immigration controls, forgetting (or ignoring) the devastating effect that immigration controls can have on the lives of so many people.
That's not to say that there should be completely open borders everywhere - you've correctly pointed out some of reasons why - if open borders leads to mass migration - it can cause other problems - so I would say there does need to be a balance struck between the needs of individuals and the needs of the wider communities. I think though that your post is wrong to the extent that it implies that the only justification for open borders is this idea of cheap labour - that's just not true. The argument about basic freedom is just as important (and I'm fairly sure it's a very significant rationale behind the EU's strong commitment to open borders).