Infinite library of videotapes is the best possible scenario that assumes existence of an observer. For a positivist this is the requirement, if there is no observer the theory is false right from the start, one doesn't have to prove it.
Once the first two scenarios are excluded, the third one is automatically true.
Nonsense. A "proof" so unwieldy as to be unverifiable is of no more value than not having the "proof" at all.
One estimate for the age of the universe is 13.772 billion years.
Since you went old school and opted for "videotapes," we'll use standard VHS tapes.
We'll use the estimated mailing weight of 7.4 ox for a VHS cassette in a cardboard case.
Since most of this video will be rather dull, image quality isn't much of an issue, so we'll assume Extended Play (6 hours per tape) for most of recorded history (get it, since we're recording everything it's all recorded history!) with about 100 years worth of Standard Play tape (2 hours per tape)
For simplicity sake we'll assume only one camera and no overlap between tapes. We'll also assume a magical camera that can switch tapes without missing any time. We'll also skip leap ears, just because this is silly enough without them.
Lets start with the 100 years of interesting stuff. There's 8,760 hours in a year, so 100 years would be 876,000 hours. That's 438,000 SP tapes. That comes out to 3,241,200 ounces, or 202,575 pounds. This means to record JUST the 100 years of interesting bits, we'd need 101.2875 US tons of VHS tape.
I was going to go ahead and do the calculation for recording the rest of the history of the universe on Extended Play tapes, but I think we can already see that the "proof" being suggested is a useless pile of plastic and magnetic tape that can't be realistically verified.