It seems there is some confusion with the terms.
Likelihood is a function of parameters. \widehat{\mathrm{elpd}}_{\mathrm{loo},i} are log predictive densities (or log predictive probabilities).
Eqs 22 and 23 do not include estimation errors for individual \widehat{\mathrm{elpd}}_{\mathrm{loo},i}. We do mention in the paragraph starting at the bottom of page 19 and continuing at the top of page 20, that we could compute the Monte Carlo estimation errors for individual terms, too, and loo >2.0.0 is computing that, too using the equations described in https://arxiv.org/abs/1507.02646. These Monte Carlo estimation errors for individual terms are often negligible compared to the uncertainty of not knowing the future data distribution.
Terms \widehat{\mathrm{elpd}}_{\mathrm{loo},i} are assumed to be independent, although they are not. This is mentioned in the last paragraph of section 5.1. With large n and well specified models the error from independence assumption is small.
The sd we report (computed as eqs 22 an 23) is the sd of a distribution describing about our uncertainty what would be the predictive performance for a new observation coming from the same distribution as the observed data. In M-open we don’t trust any model to present the future data distribution and thus we use the observed data as pseudo Monte Carlo draws from that distribution without further model assumptions. Each \widehat{\mathrm{elpd}}_{\mathrm{loo},i} presents a potential future outcome when using the model to make prediction and we would observe y_i (and leaving that observation out of posterior computation to not double use the data). Eq 23 is usual Monte Carlo error estimate for the expectation over the future data distribution.
Sorry, I didn’t understand what you try to say in this part, but maybe it’s due to the confusion on the terms?