My push back isn't really on the possibility, it's on the irony. Self hosting is for many an ideological act that's about reducing dependencies on big tech, removing surveillance, etc. LLMs are essentially the antithesis of this.
If you're self hosting for other reasons then that's fine. I self host media for various reasons, but I also give all my email/calendar/docs/photos over to a big tech company because I'm not motivated by that aspect.
Kind of but I don't really agree. Before LLMs you were still reliant on online resources, forums, digitalocean blog posts. The server itself also doesn't rely on an LLM. If one goes down, your server will continue functioning. You are also not tied to any particular LLM and can freely switch.
They also aren't seeing any of your sensitive data being hosted on the server. At least the way I use them is getting suggestions for what software and configs I should go with, and then I do the actual doing. Which means I'm becoming independently more capable than I was before.