I've had a couple of cases of VA being stuck at the start of Install mode after download. (These were during installation of Syz Dim Strings III and Syz Dim Strings Bonus libraries; I haven't yet stopped to look further into it.)
The way I got through this was to click the "X" (Cancel) icon in VA to the right of the progress status line of a 'stuck' library, then click "Install" again in VA for that cancelled installation. VA then quickly ran through the already downloaded library without any downloading happening (presumably re-verifying the already downloaded files), but towards the end of this progress, downloading automatically started up and continued until the progress indicator showed 100% done. Then Install automatically began and ran through to completion without a hitch. Download and Install were on different drives, both SSDs and having plenty of empty space.
I'm guessing that some sort of download error may have been encountered during the first pass; then presumably, VA's exception-handling code correctly prevented Install from running (although Install mode was shown as entered), but alas, didn't display any error notification in VA, nor did it manage downloading in order to attempt a fix (until the second pass, manually started after Cancel). But all this is pure guesswork on my part - hope I'm not doing VA code any grave injustice.