This really seems like a case where the existing categories in the law created hundreds of years ago are no longer adequate to capture present realities, similar to how they didn't have AR-15s in mind when they wrote 2A. If you ask whether Reddit is a public or private company, sure, it's private. But if you look at what intuitions back then were around "public" and "private", and for that matter around what "speech" was, it seems like the real answer is that (in regards to this debate) Reddit is a third type of thing that doesn't have a name yet.
> Get a blog! The very worst people on Earth have managed to keep blogs running.
This might illustrate my point better. I assume you don't support people getting banned straight off the internet. Is it simply because no one owns the internet, but Reddit is owned by Reddit (the company)? Maybe it's time for that boundary to be reexamined.