Sure Sir Barton didn't get that many blacktype but he did get a G1 winner in Easter Stockings, KY Oaks winner and Champion 3yo filly. Most stallions can't manage a GSW much less a G1 much less the Oaks or a year end championship.
Man O' War wasn't even Derby entered but it'd be a joke to think he'd have lost it and he blew them away in the Preakness and Belmont (and raced in between). He crushed Paul Jones and (with numerous upset by Upset excuses) everybody else for that matter, including Upset, and sired multiple Derby winners including until recently the latest gelding winner Clyde Van Dusen, a top steeplechaser-Grand National winner Battleship, and a TC winner in War Admiral. His direct male descendant of note lately, Tiznow-2 time Breeders Cup Classic winner. War Relic and descendant In Reality are very important in modern pedigrees as well.
Gallant Fox sired a TC winner, Omaha, and Granville, Belmont and Travers (and many other top races) winner, champion, and Hall of Famer. 2 other G1 winners (one a hurdle race, the other's included a win in the Ascot Gold Cup)
Omaha had 9 blacktype (admittedly not the most stellar) but one was 3rd in the Belmont (G1), another raced out of the US 98 times (a feat in itself but not the only one to race A LOT), others raced 112, 145, 174 (Prevaricator-5 G2s, 1 G1, and a breakdown of 24-26-29 in the money with a 10 yr career), and 115 (34 wins, and a G3) times. Seriously people, that's some serious genetic uber-achievement for a dud at stud.
War Admiral--9 G1 winners, he comes up in pedigrees all the time to this day, and the G1W included Busanda and Busher. 46,80, 65, 48, 60 are the starts of some of them.
Whirlaway had 1 G1 winner, she won the Coaching Club American Oaks and Black Eyed Susan (formerly Pimlico Oaks) and ran 26 times, and the other (Spur On, somehow accredited as a G1W) actually beat the boys in the Oaklawn Handicap and ran 77 times).
Count Fleet is definitely in many pedigrees that I've come across. Reigh Count had some serious genes. 5 G1W, the # of starts for them: 26,45,21,29, and 23. (Another raced in Ireland and is called a 'decent' racer and marked as a what I guess is group I winner but who knows)
Assault was considered sterile but pasture bred some quarter horse mares.
Citation sadly faded in pedigrees but had 2 G1W including Preakness winner Fabius (65 starts and himself sire of 7 blacktype including horses who started 64,139,56,94,41 times and 4 multiple GSWs with a G1W) and top mare Silver Spoon (27 starts) who was co-champion 3yo filly, a Hall of Famer, and only Winning Colors has done what SS did in winning both the SA Oaks and SA Derby.
We won't even go into the near misses and top sires Native and Northern Dancer. And nevermind Sunday Silence's pretty respectable TC performances. Seriously forget I mentioned them
Secretariat with his 11 G1W included Risen Star (Preakness/Belmont winner and champ 3yo male), Travers and Hopeful winner and Derby 2nd-placer General Assembly, and Horse of the Year and Hall of Famer Lady's Secret, and serious influence as a broodmare sire.
Seattle Slew had an astounding 33 G1W. Included--Landaluce, Seaside Attraction, Lakeway, Vindication, Swale, Surfside, and Honest Lady to name a few.
Affirmed had 17 G1 winners, pretty high on the list really. Includes Affirmed Success, Affluent, Flawlessly, Mossflower, The Tin Man, Peteski, and Quiet Resolve. Pops up in many female families. And Alydar's persistent 2nd place in all 3 races shows barring Affirmed's presence, he'd have run off with the thing and been similar to Seattle Slew in his TC sire influence so he's an almost-top-TC-winning-sire
See? With the exception of Assault, they all made contributions. Whirlaway had a granddaughter who was 3rd in the Belmont from his G1W daughter Scattered and I don't have the time (oh the irony) to trace the other blacktype descendants for him up into the 80s that I've found. The only thing we're calling any of them mediocre for, really, is that they didn't reproduce themselves. Oh the crime! the humanity! I'd say the percentage of quality runners, compared to your average stallion, is far and beyond what most will ever achieve. Nearly all of them received national success, not just regional, and on a graded stakes level. Anybody pointing to the ineffectiveness of Spectacular Bid, Charismatic, Real Quiet, War Emblem, etc. should know that a)they didn't win the TC, that's the point and b)several that almost did went on to influential sire careers. Overall, TC or near-TC winners have pretty big results. True duds are actually the exception, not the rule. It's just a matter of how you define patience and expectation.
