Here is a product that came together surprisingly quickly, spurred by a forum comment about Colorform shapes. I've created a sheet of 25 standard overlay patterns printed on clear plastic, with post-it style adhesive backing.
These overlays still need to be cut out, but because it's clear plastic, the cutting doesn't need to be exact - in fact, you can avoid inside-corner cutting altogether. When placed on a board, there is slight edge bubbling, but this aids in removing the stickers at the end of the game.
Aligning the overlays on the boards is very easy, due to a "happy accident" with the first test sample. The center dots of the hexes unexpectedly did not print, leaving a clear hole instead of a white dot. At first I was annoyed, but soon realized that the clear holes allow quick and fairly accurate alignment with the hex dots on the boards. This also means that LOS checks can be made using the true center of the hex.
The adhesive is basic post-it strength. I left some overlays on a board for several days, and the overlays came off without problems. The board showed no discolouration or other kind of marring to the surface - as if an overlay had never been placed there at all.
This is not a cheap product. The most cost-effective and comprehensive source for overlays is clearly the official Overlay Bundle. Self-stick overlays are a niche, 'boutique', or 'gourmet' option which will likely market for up to US$7 per sheet + shipping (still working on the sums to see if it can be lower).
As printed on the sheet:
Cut out and reapplied to the sheet.
On a board: