Your thread title does not include the special case where there is also a download version. You are suggesting this kind of homepage hosted web games discoverablilty for all projects.
I covered both cases in my reply.
I have seen attempts of having a placeholder project to drive people to a 3rd party service. Some of those have been crypto projects. So, yes, if someone would have a project that consists only of a web "game" that directs me to a another service, I would get very suspicous. I am browsing Itch and not a general software catalogue.
The "Play in Browser" filter can be understood as "Play in Browser on Itch and in the Itch app". A version hosted on some homepage would not run in the Itch app, nor on Itch. It is also understood as being hosted somehwere and not be a browser version for download. Though I would imagine the option to select "browser" as a platform for downloadable files would be usefull - but those files should not make projects appear in the "Play in Browser" category.
But this is my opinion and this is a discussion board.
For your case specifically, what kind of webhoster app would be compatible to self host your project? You mentioned special configuration being necessary. Or did you mean a browser version with "self host"? If you use the word host, you need a webserver even if it runs on localhost, otherwise it is not hosting. It is just running a browser version diretly from the file location. And if that is possible, I am kinda curious why the app would not run as a web version on Itch. I assumed some kind of multiplayer issues or whatever. But then I realized your project is a tool and seems to be made in godot and there is plenty of godot web projects https://itch.io/tools/made-with-godot/platform-web