Horse Games

Skyrim have horses too

There's a horse steam tag. Doesn't seem so hard to see what kind of people are in the screenshots.
https://store.steampowered.com/tags/en/Horses/

RDR1

How about horsepower? I can think of a few games...

Mount & Blade II: Bannerlords

Manor Lords

In Metal Gear Solid V you play as a dude and get to summon and ride a horse, but rather than the gameplay being entirely focused on that aspect, you do this mostly to travel across longer distances

Witcher 3 :o)

Your request is difficult.

Farming Simulator 22 has horses.

breath of the wild

I used to enjoy Horse Racing Manager 2 by Cyanide.

This kind of thing? https://store.steampowered.com/app/45100/Secret_of_the_Magic_Crystals/
It's an older game but I know some people enjoyed if.

There's also:
https://store.steampowered.com/app/1166860/Rival_Stars_Horse_Racing_Desktop_Edition/
https://store.steampowered.com/app/1474740/Horse_Tales_Emerald_Valley_Ranch/

I took a litte scrollstroll through steam, maybe something catches your attention :)

If you want to be a horse (though it looks a little bit cutesy):
https://store.steampowered.com/app/764240/Horse_Paradise__My_Dream_Ranch/

This one looks promising (I guess^^), but you are absolutely correct that most games like this seem to only focus on a female player character. I would assume you can customize that, but why not show it in a trailer? So this one might not be for you, but perhaps someone else is looking, too
https://store.steampowered.com/app/1559600/The_Ranch_of_Rivershine/

Not yet released, but you might keep an eye on it? https://store.steampowered.com/app/2965400/EQUIRE/
and same https://store.steampowered.com/app/3023600/Horse_Farm_Simulator_2024/
and another one, though I guess horses aren't the main focus https://store.steampowered.com/app/1151380/Cowboy_Life_Simulator/

Not too sure about that one, but perhaps? https://store.steampowered.com/app/307770/Wildlife_Park_2__Horses/

Edit: another one yet to come https://store.steampowered.com/app/2391610/HORSE_LIFE_find_horses_in_open_world_survive_in_wild_nature_as_a_foal_or_pony/

And another potential option, though I don't know about the customisability of the PC.
https://store.steampowered.com/app/276340/My_Riding_Stables_Life_with_Horses/

There is also this one released this month:
https://store.steampowered.com/app/2376970/My_Horse_Bonded_Spirits/

I'd recommend looking at the Horse tag on Steam and finding a popular one with a good number of reviews than looking for gameplay on youtube.
Chances are you'll find a few channels with people who are similarly interested to find more games.
After that I can't guarantee you will find games that "lets you be a man" but it'll be less like looking for a needle in a haystack.

I'm not a big horse person, but I bought this game for my daughters friend who is really into horses and she enjoyed it.

https://store.steampowered.com/app/1474740/Horse_Tales_Emerald_Valley_Ranch/

Not sure what age range you're after, they're 10

What about Rustler?
It is called as Grand theft horse, but I don't expect much of horse bonding.

There is also Sims 4 Expansion:
https://store.steampowered.com/app/1990600/The_Sims_4_Horse_Ranch_Expansion_Pack/

RDR2
Witcher 3

Giddyup!

I have no idea how good these games are but I can't think of anything else except this and this

Maybe these -> https://store.steampowered.com/app/1166860/Rival_Stars_Horse_Racing_Desktop_Edition/
https://store.steampowered.com/app/2376970/My_Horse_Bonded_Spirits/

Not my cup of tea, so can't say if there are good or bad ones ...

Lots of horses in Total War games