nobodywasishere Hacker News was just an example; substitute it instead for a website that does not have a snap.
Bangs and snaps are not the same, so making a custom bang does not help here. And even if it did, adding a custom one for a site individually does not help as an alternative to "site:" generally.
If you wanted to search for the user @username@mastodon.social within the website mastodon.social, you could use quotation marks, I assume. Something like searching "@username@mastodon.social" @mastodon.social.