What's the "proof" that jackknife is useful for estimating larger population? What's the "proof" that jackknife is useful for estimating larger population?
That is, since it's proposed as an estimate for a larger population, then why would one use it rather than remain limited to the sample size? Because to "extrapolate" would require that one has some sense of how much error there will be in that extrapolation and the "population total" may well be worse in accuracy than what's observed in the sample size.
https://en.wikipedia.org/wiki/Jackknife_resampling

These n estimates form an estimate of the distribution of the sample
statistic if it were computed over a large number of samples.

So why is this "extrapolation" to a large number of samples better than limiting oneself to just the sample size one has?

The jackknife is used primarily to estimate variance and bias.  It is not used primarily to estimate central tendency.  The jackknife extrapolates no more than any estimator.  The jackknife estimates the likely future performance of a procedure on the same data generating mechanism used to create the sample you're analyzing, e.g., performance on the population from which the sample is drawn.  The jackknife doesn't extrapolate any differently than other estimators, and like them does not create new samples out of thin air.  The jackknife is limited to your sample size.
