Horse Runner DX is an in-development single-button infinite(?) runner. Which I would've known from the outset if I spent a little more time reading the Steam page. Instead, I spend my time searching for additional controls in a one-button game.

On Steam: https://store.steampowered.com/app/2955320/Horse_Runner_DX/

Addendum: It seems that the developer is looking to address mass horse death, my biggest frustration with the game, so the future looks bright for horse-kind! (So long as I don't have the reins)