I am not sure if it is a bug per se.
library(srvyr)
repex_df <- data.frame(group = c(rep("a", 5), rep("b",3)),
variable_x = c(runif(5), rep(NA, 3)))
data_survey_design <- srvyr::as_survey(repex_df)
data_survey_design |>
summarise(survey_median(variable_x, na.rm = T))

data_survey_design |>
summarise(survey_mean(variable_x, na.rm = T))

data_survey_design |>
summarise(stat = survey_median(variable_x, na.rm = T))
