I would have felt less offended about it all if they'd even offered a loan handset, for the time it was to be away, but they didn't.
IIRC there's a plugin for Skype called Pamela that allows you to record the call.
One possible reaction would be to use it. Give them one last chance by calling them. Record said call, give them a liberal amount of rope to hang themselves with and if - as will probably happen - you don't get what you want you can include the entire conversation with them as an attachment in the email to the CEO (plus you can also use it to publicly shame them on Twitter if you're so inclined).
Of course doing that without telling them that the call has been recorded might be legally dubious. IANAL, so be careful.
I mean, that must be the easiest and most common repair ever, surely?
Not necessarily. If memory serves shortages in screens has been an issue over recent years, so it would only be an easy repair in the sense that it would be in the terms of the process of swapping the broken one with the new one, not the process of obtaining the new one in the first place.