Some cards that will make the list once I find time to test. Or maybe none of them will be tested. I don't know man.
Horses of the Bruinen probably should be one less mana and an Instant. But it does say the word "horse".
Freed from the Real is Pemmin's Aura #2. However it's much worse and there's enough tutors in the deck that a second copy isn't as important.
Takenuma, Abandoned Mire. Self mill is really good in an Intuition combo deck and I can re-use a Capashen Unicorn or something.
Misdirection is a second Force of Will, kind of. Force of Negation is also pretty decent. Then there's stuff like Flusterstorm, Miscast etc.
Reins of Power - Takes someone with a winning board state and turns it against them. Also horse themed.
Clutch of the Undercity - Tutors for Hippocamp and Splinter Twin or can be removal.
Thundering Mightmare - While this is not an aggro deck playing this early can quickly turn even something like a Zodiac Horse into a dangerous threat, or allow the commander to swing for lethal.
Rolling Earthquake - Reasonable board wipe and as a bonus has the word "Horse" in it.
Phantom Steed - Interesting ability but little to no synergy.
Trenching Steed - One of the worst cards in the game. I'm sensing a pattern here.
Jandor's Saddlebags and Saddle of the Cavalier are garbage yet horse themed cards that do nothing useful.
Polymorphist's Jest - Art has a bunch of horses looking at things that were turned to frogs.
Ride Down - Art is some guy getting crushed by a horse. Pretty useless for the deck. But horse.
Force Away - Horse art. Kind of bad unless I can cast my commander to activate Ferocious.
Spontaneous Combustion - Exploding Horse!
Nexus of Fate - After the LED combo erases my library if for some reason I can't win I can use Nexus of Fate to get infinite turns. Timbermare can tap all blockers so with infinite turns I should win, eventually. It also gives me a second chance should someone have a Spore Frog or something to stop the horsening. Technically you can just combo Nexus of Fate with Timbermare to win and skip the Intruder Alarm + Splinter Twin step entirely as with enough turns and no blockers even a 1/1 will kill everyone.
More tutors - Lim-Dul's Vault / Grim Tutor / Cruel Tutor / Solve the Equation. Since the deck generally just wins with Intuition in hand going all-in on tutors / searches might be worth it, even at the cost of some counter / removal cards. Honestly the deck is so dependent on tutoring Intuition that there's probably no number of tutors that's too many. Even Beseech the Queen, Dimir Machinations, Mausoleum Secrets, Praetor's Grasp, and Bring to Light. Also Tainted Pact can be good.
Wheel / loot effects - Horses are so incredibly bad. Sometimes I just get flooded with them and need to flush my hand, or I cast a bunch early game and they just sit there waiting for a wrath / chump block. I could cut down on countermagic and just add a bunch of cheap Wheels. There's also cards like Burning Inquiry that simultaneously help me dig for a win con while filling my graveyard with Breach fodder. In particular Notion Rain and Mission Briefing are all pretty good.
Additionally my horse racism might have to allow a few more horse-like but still non-horses into the deck in support roles. Non horses like Pegasus, Unicorns etc. Not centaurs though fuck them.
Starnheim Courser makes most of my wincons a little cheaper.
Cloudsteel Kirin - Robot platinum angel spirit horse. Could be niche useful which is a nice change from how bad the average horse is.
Boreas Charger - Significantly worse in practice than you'd think.
Plated Pegasus / Benevolent Unicorn - Slightly protects from burn.
Hinata, Dawn-Crowned - Reduces the cost of my stuff and makes the opponent's cost a little more.
Lu Xun, Scholar General - Teaches my horses to ride horses. Meme-y but not actually useful.
Spitemare - Bad but can be a reasonable blocker.
Lastly there's some alternative commander options. Any of them work however you lose out on Pemmin's Aura + Ghitu War Cry combat damage wins.
Princess Twilight Sparkle - Cheaper than Jegantha and buffs horses. Someone less useful abilities than Jeg's mana burst but I do like that it only costs 2 cmc to get a body in play. She also lets you play cards like Fierce Guardianship more reliably so she's probably the stronger commander overall. It is silver border though and it might lead people to believe that you jerk off to child horses.
Ersta, Friend to All - Mystical Tutor in the command zone. Shame this isn't a "real" card because it would be the best commander I could hope for.
Morophon, the Boundless - Technically a horse and makes horses cheaper. But by the time you hit 7 mana you've already played them all or won the game. Also I'm not a fan of changelings as a stand in for a tribal creature. Hilariously this combos with Dwarven Pony to 4HKO a red player which is the only time in human history where Dwarven Pony could be a legitimate threat to anyone.
Horde of Notions - There's probably a horse in the art, somewhere. Has some synergy with a handful of elementals in the decks.
Hinata, Dawn-Crowned - Removing Green and Black doesn't really alter the deck's game plan so you can go all in on American horses. Just like the cowboys used to.
Also though many of them are not horses or horse-like, these cards all have horses in the art https://scryfall.com/search?q=art%3Ahorse&unique=art&as=gridℴ=name
First version of the deck found here: https://tappedout.net/mtg-decks/horse-of-notions-2/ Current version here is magnitudes more effective.