Is the mean of a kernel density estimator a valid estimator of the population mean?

Say you have an independent sample X_1, X_2, ..., X_n drawn from some population where each X_i have the same univariate density f(x). You estimate this density function using a non-parametric kernel density estimator f_kde(x).

My question is that since, f_kde(x) is an estimate of f(x), can you use the mean of f_kde(x), that is the integral of x*f_kde(x) over all x, as an estimator for the mean of the population you have sampled from?


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I am a total novice, so we will wait and see what @Dason states - but isn't this what is done in Bayesian analyses when they integrate the posterior distribution using MCMC. Though, they are focusing on the HDI - otherwise construed as the median. So if the basic rules of mean = median seem to hold in your overarching distribution - could you be satisfied with the HDI as your measure of central tendency. This is given you are able to visualize the kernel density for any irregularities (e.g., asymmetries, multi-modal).