I'm having the same problem with Big Sur. Go to Preview, click export, slide the slider bar down to decrease the size of a large file down to say, 275kb, but it still saves the file as a whopping 3mb file, rather than the smaller file. So, while the program appears to be working, it actually isn't. We're not doing anything wrong. We're not stupid. We are following the instructions.
But here's the deal. The Preview function of Big Sur doesn't play nice with .jpg. It only works when you use apples new HEIC image format. If you fire up a HEIC file with Preview in Big Sur, you can save it as a reduced sized jpg and everything works smoothly. But if you start with a jpg file, Big Sur will not work. What works for me is to edit the file while in Photo. When I click on export, Preview converts the HEIC image to a tiff file and allows you to decrease the photo size and save as a jpg. I discovered this just on a hunch when I saw in the discussion thread that it was a problem unique to Big Sur.
I'm sorry, but just about everything you posted was incorrect.
The slider in the Export section of Preview is for reduction in QUALITY, not size. It happens to make a size difference, but that's not how to reduce the size. To reduce size, open the original image, then go to Tools, Adjust Size, change the pixel dimensions, save the file. done.
To prove it works, here is the before for an image I have:
Note that it is a .jpg file. Here is the Finder information showing it as 5.1 MB:
Now I open Tools and Adjust size and get this:
As you can see, it shows the pixel dimensions. I adjusted the size by typing in where it says "4032" and replace it with "1008" and the other dimension changed automatically because they are locked together. Here is the after shot:
Note that the file is now reduced to 289KB (from 5.1 MB). And here is the Finder entry with the new file. I did rename it so I could keep the original separate from this little demonstration:
So, use Tools, Adjust size to adjust the size and it works every time. No need to go through the hoops you are using.
And for reference, I'm on Big Sur, 11.4 and Preview 11.0.