I won't wash my hands in a gross bathroom if all I did was pee - I'd rather just avoid touching anything. There's more germs on the surfaces I have to contact than I'm going to kill by washing; and if you don't get your hands very dry then washing them is typically counterproductive - soap puts bacteria to sleep but they come back happy if you give them some water to drink. And of course, air driers just blow infected air all over your hands while the push bars for paper towels are obviously wet and thus full of bacteria.

Basically, your brother is right on the math. Maybe not on the socialization, but definitely if your goal is to have the least possible number of germs on your hands, a good first step is to only wash them in rooms that are otherwise clean.

And it's fine if you disagree with me for your own self, but the reasonable range for handwashing isn't bounded on the low end where you seem to think it is.

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Is it likely that he'll grow out of this?

No, and that's super judgmental. He's making an informed decision, he may make a different one in the future, neither one will make him immature.

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I behave similarly to someone with very mild OCD. I feel a need to wash my hands after touching a garbage can, bathroom door handle, animal, or even just after stepping into a bathroom to flip a light switch.

I'm debating just keeping bleach or alcohol with me so that I can sterilize things that he touches before I touch them myself.

No, you used to behave like someone with very mild OCD. Now you're behaving like someone whose mild OCD is getting worse. If you continue to find your fear of germs interfering with your social interactions, you should consider discussing the matter with a therapist.