Say that I had a bootstrap distribution, can I return the mode as a point estimate? It's right-skewed, so the mean does not accurately summarize the distribution.
A pragmatic answer is that a meaningful mode is surprisingly hard to estimate in real data. Say we get all distinct point estimates in the bootstrap samples, then all estimated coefficients are the mode, which is probably not what we want. We could smooth the distribution, but then the number of modes is dependent on the smoothing parameter, etc. etc.
Others here can give you a technical answer as to exactly why we take the mean and not the mode. I expect it has to do with the fact that the mean is the expected value.
However, your second sentence:
It's right-skewed, so the mean does not accurately summarize the distribution.
is not really correct, although I can certainly see where you could get that idea. The problem with the mean for skewed distributions is not that it is not an accurate summary but, rather, that for most purposes it is the wrong summary - that is, it is a good estimate of the wrong thing.
The classic example is income. In most countries, income is quite skewed so, if you want a one-number summary, the median corresponds better than the mean to what most people mean by "summary"; but for different purposes, the mean could be best, or the mode, or a trimmed mean, or even no measure at all.