But how would it get stuck? Isn't there a way for the site to "clear cache and hard-reset the page" before uploading another image?
It doesn't perform a "reset," so to speak, to ensure that the information about the last uploaded image isn't accidentally lost.
Note the sequence of events. You upload an image. Information appears about how to share it. You need to save the link, otherwise if you leave the page, you will lose the information. So, it has to be through user intervention, returning to the upload page, so to speak, by doing a "refresh" or clicking on "new upload".
Since everything happens on the same page, it ends up being a protection system to prevent unintentional losses. To minimize this, without compromising security against unintentional data loss during sharing, you could configure it to open a new page with each upload. However, I think that would be less practical than clicking a button.
Furthermore, if you have multiple images to upload, you can upload them all at once.