The following JavaScript code locates the first video element on the page, removes the attributes that may be blocking the PIP feature, and attempts to open the video in picture-in-picture mode.
var v = document.getElementsByTagName("video").item(0);
v.removeAttribute("disablepictureinpicture");
v.requestPictureInPicture();
(The code does not contain any fallbacks. Either it works, or it doesn't.)
To use it, you simply need to create a bookmark in your browser and enter the following as the URL:
javascript:var v=document.getElementsByTagName("video").item(0);v.removeAttribute("disablepictureinpicture");v.requestPictureInPicture();
Now start your video, click the bookmark and enjoy your video in PIP.
By the way: This bookmark even works on Disney+!