Re: "enough grunt to take on tasks that would leave the Pi panting in exhaustion"
Wat.
Are you REALLY comparing a crappy Atom made in 2008 with one made FIVE years later, with all the IPC improvements and a fundamental restructuring of the entire architecture?
The Bay Trail one is at least 3x more powerful than the one you've got in that eeePC, and it also comes with a more advanced underlying architecture which can cut thru higher-rates 1080p with no issue.