How well-known twitch bots are deployed?

Website and bot are two very different things.

Bots are long running scripts or services.
Websites, generally, only run/exist when a user calls/loads the page.

I don’t, I get the caster to mod the bot manually.

Since theres no API for “give the user mod” you would have to get some “dangerous” permissions from the streamer that they don’t want to give out.

And if there was an API I still wouldn’t ask the caster for this permission due to the streamer generally not wanting to hand control of the channel (and it’s mods) to a bot.

What do you mean “site”?

Cheap and reliable often do not go together.

Cheap often comes with rules and restrictions.

I prefer bare metal dedicated servers. Currently using OVH