So this one's a bit tricky as I don't have screenshots. I've noticed that some of my mares, not all of them, won't have their leaderboard standings showing on their page until I click on their name directly from the board they're on and then it pops up on their page like it was always there when it wasn't prior. I'm not the only player that has noticed / had this issue

I've experienced this for a long time. I do most of my breeding mid-month and at least 50% of the time the LB designations don't show on their page, so I have to go to the regular leaderboard page to see how the LB mares moved after their most recent foal or if a new mare debuted after her 3rd foal. I've missed a few debuts from this bug if I forget to check the LBs directly.

Here is a screenshot of my LB Mare Manglar LS La Silla now showing her LB status on her page, but clearly on the LB leaderboard. I have not bred her yet or done anything to change this.

The leaderboards on the horse page update once per hour. Is it possible you bred them, then looked on the leaderboards, before the page updated?

No as I bred the mares on the first, went back to bed and when I got up they didn't have anything on their page which I thought was off since they had placed in the top 20 for their breeds LBs the previous season. After I opened their link from the LB page it immediately showed up on their page where it's supposed to.

I have screenshots of the issue now. Mare was bred on March 1st and is #25 AALB but doesn't have the LB standings on her page.

Her leaderboard is listed on her page. It is what I said before: Leaderboards update once per hour and during that time, the leaderboard placing won't show on their page. You just have to be patient and wait for their leaderboard to show up.

Even though she's been on the LB since March 1st? It only shows up after I click her name from the LB but it'll disappear again randomly until I reclick her on the LB

Go to her page at 15 minutes after the top of the hour and see if it's there. :) I think the leaderboards update around 45 after the top of the hour, so it won't show around that time.