(1) Security. An always-on, externally accessible device will always be a target for breaking in. You want the device to be bulletproof, and to have defense in depth, so that breaking into one service does not affect anything else. Something like Proxmox that works on low-end hardware and is as easy to administer as a mobile phone would do. We are somehow far from this yet. A very limited thing like a static site may be made both easy and bulletproof though.

(2) Connectivity providers should allow that. Most home routers don't get a static IP, or even a globally routable IPv4 at all. Or even a stable IPv6. This complicates the DNS setup, and without DNS such resources are basically invisible.

From the pure resilience POV, it seems more important to keep control of your domain, and have an automated way to deploy your site / app on whatever new host, which is regularly tested. Then use free or cheap DNS and VM hosting of convenience. It takes some technical chops, but can likely be simplified and made relatively error-proof with a concerted effort.