Recent foals are in the new app (thanks @willagrey for the nudge 
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When you have new foals (judged by when you last logged in, or yesterday if you’re like me and log in every 10 minutes)
Clicking on the “Your stable has x new foal” gets you to a page similar to the recent results page
Each foal is listed, with the option to rename (unless stillborn - rename in white/black indicates that the horse already has a name, if it’s in blue it’s not named). Just in case you’re scanning the list quickly for babies that need names.
Similar to recent results, the “Copy List for Forum” insta-copies the foal info for you to paste into the forum. If you do rename foals, you’ll need to refresh this page in order for the forum paste button to pick up the new name (I should be able to improve this at some point, but it’s not high priority.)
Note that this list is paginated when you scroll down, so if you want the full list to copy to the forum, you have to scroll down until the full list loads, and then the Copy button will have all the foals. (I might add the copy button to the bottom as well, thinking about it. Need to figure out how to do that only if you’ve scrolled and gotten more results. 
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Also, if you want to pretend Past You was totally on the ball and posted foals as they were born, you can use the buttons saying “x foals were born on <date>” buttons, which will just give you the list on that date. You can tell because the URL has “&this_date=true” in it. If you just want all the foals born since that date, remove that from the URL and you’ll get all foals owned/leased by your stable that have been born since that date. (Yes, in theory you could put in 1990-01-01 and it will load every single horse you own/lease and their breeding info. Please don’t. )
Note that the only difference from old app is that it doesn’t load 3/4 siblings, because that logic was super gnarly. I may add it back in at some point, probably when I re-create the hypothetical mating page as it was used there. It does do full/half sibs, listing them if they’re stakes-quality.
Shout if there are any questions/issues.