No need to define a custom function, instead we could simply sum all elements of the group, subtract the current value, and divide by number of elements per group minus 1
.
df %>% group_by(grouping) %>%
mutate(special_mean = (sum(value) - value)/(n()-1))
# grouping value special_mean
# (chr) (int) (dbl)
#1 A 1 8.5
#2 A 6 6.0
#3 A 11 3.5
#4 B 2 9.5
#5 B 7 7.0