Are you letting the foam part completely air dry for a day or two before putting it back on? If so, then it's just a bad design - that's a really small foam piece that I don't see in uprights. I have a couple hepa vacuums with large flat foam pieces along with a suction indicator. If it's near red, it's time to wash out the foam, but we haven't had to do that in quite a while.
As far as what would happen if you remove the foam piece - your hepa filter would be eating all the large dust particles too and so it would become a lot less effective on the small ones (what it's designed for).