Push costs pennies. It's an arbitrary restriction.

If you want to run your own push for pennies all you have to do is compile the client yourself.

I'm not going to recompile and redistribute a binary outside the Play Store.

Then your piggybacking on their infrastructure. I don't think they are unreasonable. "It can be done for pennies, but I won't" sort of implies that it does indeed take more than pennies worth of effort.

Then you pay for it. Nothing stops you

That's precisely my point. It's an arbitrary rent-seeking restriction.

Publishing an app in popular app stores, for an organization, requires several $100 in annual fees. That’s before any mobile app is even published.

At this point I think he's just trolling. Nobody can be this entitled

It's a yearly fee that amounts to a couple hundred dollars. That's about an hour of an engineer's salary. Zulip's customers make this less than a fraction of a fraction of a fraction of a rounding error.