So, this is a double-pronged bug report, but I didn't want to start two whole new threads for what is essentially probably coming down to the same piece of code in the backend.
So, first off, let's talk about the scheme in the URL, or rather, the lack thereof. I think it is either a bug or a misguided feature that profile links do not show the scheme. While `https://` is going to be the most common scheme by far, I actually have a valid use case for using `http://` in my profile link (I bought a domain purely to register redirects to my socials and such, and some browsers seem to not like it if I use `https://` with those links, so I use `http://`instead.) However, it's completely reasonable for users to expect secure connections, and while most browsers have some sort of built-in indicator that warns you if you're browsing a site that doesn't have a secure connection, they don't always warn you before you actually first load the page. Users should have a right to be able to anticipate an insecure connection, so the best way to make sure it's always telegraphed is to just display the scheme. I did not want to make this a feature request because I believe it's kind of a given, and because the next part of my post is something that I believe is being done intentionally but which actually causes bugs.
Secondly, the extraneous `/` suffix that gets added: I'm not an expert with web stuff, but doesn't it add the risk of changing the meaning of the link? Also, it's an eyesore when there is no scheme displayed. Anyway, let's give examples of what behavior I am talking about: if I type `http://drewbabe.net` into the link page, it gets rendered as `drewbabe.net/` and the href attribute is `http://drewbabe.net/`. That's not what I want, I want what I typed to be what shows up. And I don't think it's just a me problem, since you can even see this on the @staff account, which shows their link as `assc.notion.site/` with the href attribute being actually `https://assc.notion.site/`. Again, I'm not an expert with web stuff but it does impact the meaning of URLs apparently, so this may cause unexpected and unavoidable behavior for some people. This applies especially if you ever start letting people use links that aren't HTTP(S) (thinking primarily of ipfs:// URLs, which, if you have a local gateway running, are more secure than going through a third party's HTTPS gateway.)
All in all, I would like to suggest that the behavior the user should experience should be something like this: