I'm up for that. It'll take a while to download all the source videos, but I'll report back with results and videos when I'm done.
I already did a test encode with x264 --preset veryslow --tune film --level 4.0 --crf 21 on the Bourne Identity clip, and the result is within the same realm than the Beamr video with ~58% the filesize. Which is not surprising at all, considering Beamr's settings.
EDIT: Testing with nearly identical settings (x264 --bframes 0 -- subme 8 --no-psy --no-mbtree) using the latest x264 gives me about 40% the bitrate of the Beamr example for the Bourne Identity clip, which unsurprisingly looks notably worse. So maybe Beamr's "magical" encoding technology is all about fiddling with x264's CRF to make it double the bitrate at a given value compared to vanilla x264?