Like others have said, this has to be handled carefully and slowly, and the degree of mental illness matters tremendously. But generally speaking, long-term, you will always be better off building trust with these relatives rather than breaking trust by getting rid of their things behind their backs. As we age there are fewer and fewer things parts of our life we can control, and that is scary. Personal belongings are one of those remaining centers of control. So do your best to be compassionate, even when you want to rip your skin off in frustration at what seems to you like easy decisions to make.
Crap is in the eye of the beholder. Pretty much by definition, a person's belongings are more valuable to that person that to anybody else, including relatives, looking in from the outside. I admit I actually laughed a bit at the opening of your post where you describe your own collections by saying they "are not in the way" and "if anything, these enhance the appearance of our house," because that is EXACTLY the sort of thing I would hear when trying to help certain elderly relatives deal with their own abundance of stuff.
Everyone's situation is different, but I hope the following example will be useful to you. With my relatives, I found that by going really slowly with them in the beginning, I was able to establish trust that eventually sped up their clutter decision making. At the start of this process, one day I suggested to a relative that perhaps we could free up some room in her closet by donating some of her shoes to a charity she supported, and could I help her with that. Then, while she sat in a chair watching me, I spent the afternoon pulling out and arranging her shoes across most of her living room floor, grouping the shoes in clusters (like "black heels" and "tennis shoes") so she could see in one glance what she had. Then I asked about each cluster of shoes, saying "which of these would you like to keep?" To me they all looked like duplicates, but they were her shoes, and therefore the choice was hers to make. She was surprised by how much she had, and ended up getting rid of half of them. But even if she'd gotten rid of nothing that day, I still would have accomplished my real goal of showing that I could be useful in helping her deal with her stuff in a non-judgemental way. After a few similar exercises, we eventually reached the point where I could ask her more directly about individual items, and she'd make a call instantly.