For sites that have a built-in download function, the process is easy enough - just start downloading a video and, while it's downloading, open the still-downloading file in MPC-HC, mpv, vlc, etc - it will play back without issue as long as you don't try to jump ahead into part of the video that has not actually downloaded yet (also note that AVI files can have issues seeking altogether), though you need to make sure you select the correct file as many times you'll actually get two files while it's in the middle of downloading - one of which is a 0KB file (which is not the file you want to open in your media player, open the other file).
Alternatively you can copy and paste the download link and try opening that directly in your media player of choice, but this can sometimes be a bit more problematic.
There are some browser extensions out there for playing video files and/or URLs in a media player, I personally use "mpv-youtube-dl-binding" which adds a "play with mpv" option to context menu for right-clicking on HTML5 videos (though it does not automatically pause a playing video nor stop an already-downloading video), and it actually seems to work with other media players as well like MPC-HC v1.8.x when configured to point to that program's EXE instead.
For sites that don't have download functions, you can still usually use SVP if it's a site that can use non-DASH/MSE HTML5 video playback by simply right-clicking the video and then selecting the according option to download the video or to copy the video location URL. Note that for some sites, you need to hold down the shift key while right-clicking on a video in order to get the normal right-click menu (this is not supported in all browsers however, and I've only tested it in the browser I personally use - Pale Moon, though it theoretically should also work in Firefox and any of its other forks/derivatives).
This method may also require you to force the site to not use DASH/MSE; I know Pale Moon has a setting right in the options to disable DASH/MSE without disabling HTML5 video, but I've no idea about other browsers (heck there's even a Pale Moon extension that lets you enable/disable DASH/MSE on a per-site basis).
Additionally, on some sites you can't even download the video file via right-clicking and instead need to find it via something like "Page Info" -> "Media" (protip, if you then click on the "Type" heading twice, you'll get videos sorted near the top) and then click the "Save as" button or copy & paste the file URL, note that the file extension can sometimes be .htm or .html even if it's still actually a video file (they'll still play back in MPC-HC, VLC, mpv, etc without issue). I know this "Page Info" function existed in Firefox back in like version 24 but I've no idea if modern versions still have this "Page Info" function, I just know that it exists in Pale Moon.