Seriously, when was the last time your CPU died? Last time for me was years ago, when one of my Athlon MPs fans got stuck and it overheated. That's because Athlon XP/MP had no thermal protection. All modern AMD CPUs do, and Intels have had it since the Pentium Pro days (if not earlier, but I had our PPro overheat because of a defective fan, and after replacing the fan it worked fine again.
As for the fileserver/Media PC example, I wonder why you'd ever want to shell out $300 for such a thing. The Atom offerings are more than enough for doing that (For a mere fileserver, it's overkill: I use a Soekris net5501-70 for that.) Take an Atom 330 with an ION chipset or go for an Atom D5xx (You'll have to look, not all come with DVI/HDMI) and you're set for $150 given your requirements. If it breaks, and it won't unless you have a lemon, replacing it won't be expensive at all.
Oh, and in your scenario, you could just as well buy an older CPU for very cheap. CPUs for older sockets are still being sold. You want a 775-socket CPU? No problem! It's been out since July 2006 and you can still buy CPUs and they're cheap. So for just repairing the situation, shell out the minimum and be happy again for the next 3 years... until the CPU burns out again (unlikely). In that time, yet aside $10 per month for a future replacement, and you'll have a grand total of $360 to spend on new gear...
The problem with your scenario is that you want to use the "broken PC" situation as an excuse to upgrade. However, from the scenario, what is important is that you get the machine back up and running, ASAP so that the incoming visit can watch movies with you. Get your priorities right, and the scenario has only one valid solution: replace CPU with cheap replacement and get on with business.