So, yeah, sampling would be a mostly futile effort since you're looking to estimate about 8 to 10 decimal digits of precision. Though it's technically still possible since you'd expect about 1 in every 50 million - 5 billion IDs to work (assuming somewhere between a trillion and 10 billion videos).
My statistics knowledge is rusty, but I guess if you could sample, say, 50 billion urls you could actually make a very coarse estimate with a reasonable confidence level. That's a lot but, ignoring rate limits, well within the range of usual web-scale stuff.