I can't find Airplay on my Control Center and on the Share menu in the browser

that is because AirPlay isn't a Control Center option.. and Airplay also isn't something available in Safari share menu

One thing you can do in Control Center is use the Screen Mirror feature, which is discussed in the article you came here from.

Using that, you could screen mirror while you are using Safari? Might help to tell us what it is you are trying to achieve.

Or are you thinking of AirDrop? You could share a web page using AirDrop...