"None of the solutions offered by MS have worked and they have helpfully suggested I backup my personal data and then nuke the machine and start again..."
Oh pshaw. Just call them back about an hour later, and say, "Thanks for the advice, Backed up my data, nuked the system, and installed Linux Mint. It works great now!"
Then hang up.
(BTW, don't really have to install Linux, just say you did.) ☺