Happy to report that I successfully replaced the glass & digitizer on my Nokia Lumia 800, thanks to a very informative video on YouTube and a friendly parts vendor in the UK.
The repair was just about as easy as promised: two small screws to release the old glass and two small connectors to connect the new one to the phone. Here’s the video in case you want to try this yourself:
It took a week or so for the replacement part to arrive from the UK (what I get for owning a European phone in the US), and I was happy to finally get it. In the meantime, I was starting to phone it old school: