I would argue not from the perspective of "asymptotically, the empirical distribution will be close to the actual distribution" (which, of course, is very true), but from a "long run perspective". In other words, in any particular case, the empirical distribution derived by bootstrapping will be off (sometimes shifted too far this way, sometimes shifted too far that way, sometimes too skewed this way, sometimes too skewed that way), but **on average** it will be a good approximation to the actual distribution. Similarly, your uncertainty estimates derived from the bootstrap distribution will be off in any particular case, but again, on average, they will be (approximately) right.