Look at it this way: there's a finite amount of OH radicals being produced each interval of time, which reacts with some CH4 and disappears in the reaction.

You are correct that OH radicals regenerate, but more methane => "breakdown capacity" becomes overwhelmed.

This is known as zeroth order kinetics, similar to alcohol metabolism. Drinking 2 beers = 3 hours until sobriety; 4 beers = 6 hours. Your liver has a fixed capacity, so drinking twice as much doesn't double the metabolic rate.

I'll add that you're also entirely correct: if there's some long lived chemical that catalyzes OH formation, sending that up instead might be a good remedy IMO. If there's no collateral toxicity...