What caught my eye was that in 2011 they couldn't shoot it in 6K stabilized and crop for motion afterwards. But there were already 4K cameras at the time, and with zooming they should be able to capture at least the critical time window of Beckham kicking the ball / landing stabilized for easier post-processing.

But the nature of motion glitches found suggests they did actually post-process on a shaky video. They could do a better job, but then the trick worked for more than 10 years, and thus fulfilled its purpose more than enough.