No. It does not help. What do you mean with self-hosting? It is not an unreasonable question. This very thread here comes up on page 1 or 2 when searching for its usage on Itch https://duckduckgo.com/?q=site%3Aitch.io+self-hosting&ia=web
And wether or not you are interested in my opinion about your situation, I asked, if I understood your situation correctly. Judging from your reaction, I did understand. Sorry for bringing my point of view into a situation. And to clarify my view: I fail to undertsand why you do not upload your web version to Itch for the people that do can use it here. This would also have the benefit that you do not have to wait till your suggestion is implemented.
As for the general case, if you had answered the example with the Windows version on Itch + Linux version elsewhere scenario, there would have been a base for further discussion, depending on how you would like the discovery system to work, based on platforms. I consider that scenario and the one with external web hoster quite similar.