We already have a tool for site-wide banners, as I mentioned in your other post.
In terms of sharing your community as a site or category you already have that tool available to you, although it’s surprising just how many administrators and moderators overlook them.
As we’ve acknowledged above, og:image is set to the first image of the initial post, and every category already includes an “About the XXX category” post. You could easily update these posts to serve as an initial introduction to the category including any imagery you want to appear when shared off-site and share those posts externally. They’re horribly underused by most sites, I think we’ve all developed a habit of tuning them out - when we’re setting up new communities or categories defining each can almost be an inconvenience.
Site-level promotion is equally simple, create a post and use it as your go-to point of entry for new users.
An intro post would solve this too. You can set your logo as the initial image and control exactly what others see when posted externally, although if you’re using Discourse og:image for / is going to be your logo already.