What do you mean with "measure a p-value"? You make your observation, calculate a statistic (a function of the observation), and look at the distribution of that statistic under the null. The p-value is, by definition, the percentile of the value you got in that distribution (which might or might not be the actual distribution).
You want to check if a die is loaded to yield 6 more often than it should. The null hypothesis is that the die is fair. You can calculate the distribution for the number of 6's in 3 rolls (0: 58%, 1:35%, 2: 7%, 3: 0.5%). You roll the die three times, you get three 6's. The p-value is 0.005. Do you agree? The p-value is 0.005 whether the die is fair (the null hypothesis is true) or loaded. Do you agree?
> Making future observations that are consistent with a known hypothesis is exactly what reproducibility refers to within the context of science.
Scientific experiments are usually about rejecting the null hypothesis. For example, the null hypothesis might be that there is no Higgs boson and the peak observed in the LHC data is just noise. They made their analysis and rejected the null hypothesis (p-value less than 0.000001, do you think they calculated it properly?). In this context, reproducibility means "finding the Higgs boson again if the experiment is repeated" and not "repeat the experiment and get a result consistent with the null hypothesis".
According to your description of the limits of p-value analysis, the only conclusion that physicists should get out of the experiment is that if they do it again they should expect to get results consistent with the null hypothesis (i.e. no Higgs boson) with 95% probability. But they see it as evidence that the null hypothesis is false and the Higgs boson real.