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I'd actually love to know too, been wondering the same thing :)

I was a bit skeptical about the foal PT averages, but it works for these guys in the liver club. There is a 0.01 point difference between #6 and #7, with #7 being 12.87 compared to #6 with 12.86 average foal PT. It could also be noted #7 has not had a foal this season, so a possible factor?
#10 also has a higher foal PT average than #9. #10 has not had any foals this season, while #9 has.

Of course there are probably more deciding factors, I'm just taking guesses and i had fun looking these up XD

Top Producing Stallion (Liver Breeders Club)
1. 6 Grim Reaper DPsa (av foal PT 13.05) > https://www.huntandjump.com/horse.php?horseid=11939274
2. 6 Armeniel of Samos (av foal PT 13.02) > https://www.huntandjump.com/horse.php?horseid=11565493
3. 7 Blackjack DP (av foal PT 13) > https://www.huntandjump.com/horse.php?horseid=11521848
4. D7 11289724 (av foal PT 12.91) > https://www.huntandjump.com/horse.php?horseid=11289724
5. V8 DP Smooth Sailing (av foal PT 12.9) > https://www.huntandjump.com/horse.php?horseid=11262249
6. 7 Takashi (av foal PT 12.86) > https://www.huntandjump.com/horse.php?horseid=11881051
7. 6 Black Benjen DP (av foal PT 12.87) > https://www.huntandjump.com/horse.php?horseid=10979426
8. V7 DP Unable Reduce (av foal PT 12.69) > https://www.huntandjump.com/horse.php?horseid=10728716
9. 5 Wild Alaska DP (av foal PT 12.51) > https://www.huntandjump.com/horse.php?horseid=9562466
10. 6 Katakuri II (av foal PT 12.64) > https://www.huntandjump.com/horse.php?horseid=10957969

I think they need to be living and had a live foal the last 2 breeding seasons for the older horses to be eligible for the leaderboards.

They need to have 10 living foals to have a chance for a place on the leaderboards, and the placement is based on the 25 best foals.

I don't know if it is the showing ability only, or if it is the combined breeding/showing ability that the foal are judged on

MasterStablesMS is correct.