Is it normal to think “yours’s” should work

"Yours" is a possessive pronoun, so it's not treated the same as a noun in written English. "Its", "ours" and "whose" are other possessive pronouns. It is simply a convention that the possessive apostrophe never appears in these words, although the sound in spoken speech is exactly the same as "your's", "who's", "our's" and "it's". (As I'm sure you know, "it's" and "who's" are contractions of "it is" and "who is" or "who has", so the apostrophe does serve a useful purpose in distinguishing between two different meanings.)

I haven't looked into it and I'm definitely not an expert on English grammar or etymology, but I assume that at some point there was a consensus that the apostrophe should be dropped from possessive pronouns, even though they followed the general rule that possession is indicated by a final s sound in spoken speech.

"Yours's" makes no sense whatsoever because that would indicate some sort of double possession.

The example you give of “All of these cat paw prints belong to our cats! Which of these paw prints are yours’s?” doesn't make much logical sense either, since the questioner seems to be asking which of the other women made the paw prints, not which of the cats did it. It would make more sense to ask, "Whose cat made these paw prints?"

JellyBeanBandit mentions "mine's" and gives the example of "they're mine's". I accept that this is used in colloquial English, but it sounds uneducated to me. "Whose are these cat prints? They are mine's," is comprehensible, but a more grammatically correct (and logical) response would be, "My cat made them."

Something that causes a lot of debate and confusion is the appropriate use of either a possessive apostrophe alone or an apostrophe and s in proper nouns that end in s. This is largely a matter of style, making the meaning clear and sometimes replicating what would be said in spoken English.

You are correct that you would write, “These are the mountain pass’s trails”, and you would also write, "The Brenner Pass's main road is now clear of snow." But most style guides would recommend, "The Beatles' albums," rather than, "The Beatles's albums", because if that phrase was spoken, it would not sound like, "The Beatleses albums". However, you would write, "That is the Jones's house," because when spoken, the sentence would sound like, "That is the Joneses house".

Who the fuck takes their cat to a party anyway

That’s when you just rework the sentence. Or I believe if a word ends with an s and then you add the ownership apostrophe, you don’t need the extra s (Travis’) In the case of “yours”, I believe it can be added to the list of shit in the English language that just “Is what it is”. Lol.

I lost it at they all bring their cats to party lmao

I get your logic on this, dude. It makes sense, but I think yours is self covered for the reasons Boojum mentions. There are odd ones where this comes up though like Zues's where you would be correct to double "s's".

"Which of these people prints are yours's?" I don't get that, is that a mistake? Shouldn't it be "Which of these cat prints are yours's?".

If you did mean to say "cat prints", then I get what you mean. Like if they said "Which of these cat prints are yours?", then the reply should be "They're not mine, I'm a human, I don't leave cat prints".

I'm afraid I can't explain exactly why it shouldn't be allowed, but it is an interesting question.

I think the word "mine's" sounds a lot more correct than "yours's" even though that'd be breaking the exact same rule as "yours's" (eg. if instead of replying "they're my cat's" you replied "they're mine's"). So if people brush off "yours's" but are more forgiving towards "mine's" then that is a bit inconsistent.

"yours's," try spell check!