The thing that makes casting so appealing for me is how ubiquitous it is. It eliminates situations with guests where they would recommend a show/movie only to find out that I can’t easily play the content because it’s only available on a streaming service that the guest pays for and I do not. As long as the guest brought a device and connected it to my WiFi, it more than likely could be casted without having to install another app and/or sign up for a new service (or have the guest login with their account).
I am becoming less optimistic about it though. I just do not think that the level of ubiquity that Chromecast reached even 10 years ago will be matched with a FOSS alternative. Developers would need to incorporate it into their apps, websites, etc. or it would need to be compatible with existing solutions. I doubt Google will open Chromecast up enough so other options can be fully compatible with it. Additionally, without the backing of a major corporation, I do not see developers taking the time to make their content compatible with another casting option.
The steps below are high level, but should provide an outline of how to accomplish what you’re asking for without having to associate your IP address to any domains nor publicly exposing your reverse proxy and the services behind the reverse proxy. I assume since you’re running Proxmox that you already have all necessary hardware and would be capable of completing each of the steps. There are more thorough guides available online for most of the steps if you get stuck on any of them.