First, bootstrap cannot remedy a problem of an unrepresentative original sample. Thus I agree with @CliffAB.
The chances of randomly drawing a sample that is little representative diminish as the sample size grows. If there was just one coin throw, the outcome would always be "crippled": one element (either head or tail) of the two (head, tail) that are in the population would be completely absent in the sample. Meanwhile, as @justanotherbrain notes in the question, the chances of having 10 heads out of 10 coin throws are just below $10^{-3}$ -- much better than when you have only one throw. @whuber correctly notes that bootstrap properties are asymptotic. I will add that, under certain conditions, bootstrap is more efficient than standard estimators, e.g. achieving convergence rate of $n^{3/2}$ rather than $n^{1/2}$. (I will not expand on these conditions here; a detailed treatment is given in Hall "The Bootstrap and Edgeworth Expansion" (1992) (this is a whole book).)
But let me provide another perspective why bootstrap is generally relevant. Sorry if it does not address your question directly enough.
Bootstrap is useful for assessing whether an estimator (a formula) works alright. The idea is as follows.
- Consider the original sample (of $n$ elements) as "a population"
- Draw an $n$-element bootstrap sample from the original sample
- Apply the estimator (the formula) onto the bootstrap sample to obtain its realization (a value)
- Iterate 2. and 3. many times, save the realizations of each iteration
The realizations from the many iterations will form an empirical distribution. Its characteristics (mean, variance, whatever else) will be accessible to you.
Since you consider the original sample as "a population", you know the true characteristics of that population. You can then take at the empirical distribution (based on the many bootstrap realization) from point 5. and see how it looks relative to the actual population characteristic (one item) that the estimator was supposed to estimate.
You can thus get an idea whether your estimator (a formula) is biased, how large its variance is etc. All that is valuable information when it comes to assessing properties of an estimator. Even better, the original sample should normally be fairly representative (contrary to your example) of the actual population of interest. Hence, you can extend your conclusion from the bootstrap experiment to the original problem of interest (by analogy).
For example, if you see that the estimator tends to underestimate the population characteristic by $b$ units ($b$ stands for bias), you would add $b$ to the estimate based on the original sample so as to remove the bias. That way you would get an unbiased estimate of the characteristic that you were after in the first place. Thus you would fruitfully utilize what you have learned about the estimator when bootstrapping.
But at the end I have to repeat that bootstrap is not a remedy in case of an unrepresentative original sample, as your example illustrates.