I've been having the same trouble. On a website, Chrome or Edge won't trigger the upload/attach dialog when clicking a button on a website, but drag-and-drop will usually work on sites that allow that method. Clicking on the Download button on sites also does nothing. The buttons don't even highlight or anything to show they've been clicked.
However, it seems there IS an upload/download process that has been initially triggered, but it goes nowhere. Because if I click on download buttons in Edge or Chrome, they do nothing at all. But if I then close the app, I get a warning telling me that a download is still in progress. I confirmed that there are not any partially completed or in-progress downloads in my Downloads folder. But it knows the process was attempted. This seems like a system-level problem. This is also identical to how the Save to PDF error fails: clicking on the button to do so shows absolutely NO visible response.
Also, printing to PDF from ANYWHERE in MacOS, (Monterey 12.0.1 on 2020 M1 Mac Mini), doesn't work. Workaround is sometimes available to Open PDF in Preview, and then Save it there.
Also, also, printing from Chrome is currently borked. Pages come out formatted a bit wonky. Printing from other browsers is usually better. Not sure if this is related.
Like others, I also found using Safari a useful workaround, but over time when I was up- and downloading and printing a whole bunch of documents, it stopped working too. So I switched to Firefox, and it too worked only for a while, and the same again with Edge. Updating and restarting Chrome for me fixed it a couple weeks ago, but then the problem came back.
And to be clear, updating Chrome fixed both the uploading problem AND the PDF/Printing problems. But all only temporarily. So it seems clear they're all related.
I'm going to try the solution mentioned of updating permissions for all 4 browsers and see how far that gets me. But given the systemwide related problems, it seems like an Apple problem to me, maybe regarding the M1 chips, or maybe Monterey.
UPDATE: problem persists even after updating OS to 12.2, updating Chrome to 97.0.4692.99, and updating all browsers' permissions as noted above, and doing a full system restart.
Chrome, will NOT download, upload, or print to PDF
Edge, will NOT download, upload, or print to PDF*
Safari, WILL download and upload, but will NOT print to PDF
Firefox, WILL upload and download, will NOT print to PDF
Preview, WILL print to PDF.
As before, the features above that ARE working, I would expect to eventually begin not working, based on prior experience.