Does everything you just said not also apply to fingerprints? You leave them everywhere you touch, and people touch a lot of things. Maybe less so than DNA, since it's effectively a subset (things you touch likely have DNA from your hand skin, but things you don't touch might also have DNA from other things), but it's still just saying "you were likely here". And given a large enough population, wouldn't there also be many partial fingerprint matches?

I'm not sure how these two things are issues when comparing fingerprinting to DNA matching. They seem to be applicable to both.