I'm under the impression that when you bootstrap, your final results are the original statistic from your sample data, and the standard errors from the bootstrapped trials. However, it seems more intuitive to take the mean statistic from all your trials, rather than just the statistic from the original trial. Is there some statistical intuition why it is one and not the other?

Also, I came across a use case where someone uses bootstrapping using the mean as the statistic. They did their sampling, took the mean of each trial, and used that to calculate the confidence interval around the mean. Is this ok? It seems like you could draw confidence intervals using the original data itself, and bootstrapping would artificially lower the standard errors. Again, is there some intuition I could use to understand why this is ok/not ok?