In my view, a "frequentist" would be limited by a single observation and forced to conclude that the standard error or confidence interval associated with any associated parameter was inestimable or undefined. So, while I can understand how bootstrapping would produce standard errors, I don't understand how you're able to obtain those metrics using traditional, "frequentist" models without it.
A Bayesian, on the other hand, would not be so constrained -- see chap 13 of Andrew Gelman's book Data Analysis Using Regression and Multilevel/ Hierarchical Models where he states that estimating these metrics can be done even with an n of 1. His point is that the Bayesian approach creates a wide set of information in the posterior distribution which draws on and shrinks information about these underidentified factors from the full set of data.
That said, the issue is less about the type 1 errors -- statistical significance -- than it is about power and type II errors. You can obtain a significant result in terms of a p-value while having little confidence that the finding is a real one. In other words, even with the potentially huge amount of empirical information available to the Bayesian posterior distribution, with an n of 1 your statistical power remains limited. Confirmation of this can be found in the observation that Bayesian approaches will give answers possessing a high degree of accuracy that can still be completely wrong.
Gathering data sufficient for large-scale genome mapping of rare mutations is a separate topic best dealt with by computer scientists since it can involve integrating confidential, private and encrypted information from many data sources as well as across national boundaries and continents. This is not a trivial challenge. Shafi Goldwasser, computer scientist at MIT, has a paper which won the ACM Turing Award for its development of a game theoretic approach to sharing data that solves these challenges and concerns.