Syntax for using data-masking functions in dplyr::distinct()

I am trying to merge rows that have duplicated entries in the ID column, by retrieving the means of each column.

In dplyr::distinct()'s documentation it states that distinct can take a data-masking function as input and return new values for the resulting column.

I used an example from the help pages. However, I am unable to get it to work properly, and could use some help with the syntax.

library(dplyr)

a <- data.frame(
    "ID" = rep(1:3, each = 2) %>% as.numeric(),
    "x" = rep(1:3, each = 2) %>% as.numeric(),
    "y" = 1:6 %>% as.numeric()
)

# Original dataframe
a

A data.frame: 6 × 3
ID  x   y
<dbl>   <dbl>   <dbl>
1   1   1
1   1   2
2   2   3
2   2   4
3   3   5
3   3   6

The example from the Help pages:

grouped_mean <- function(df, var, ...) {
  df %>%
    group_by(...) %>%
    summarise(mean = mean({{ var }}, na.rm = TRUE))
}

distinct(a, ID, grouped_mean(a,ID), .keep_all = TRUE)

# Output
A data.frame: 3 × 4
ID  x   y   mean
<dbl>   <dbl>   <dbl>   <dbl>
1   1   1   2
2   2   3   2
3   3   5   2

Instead of evaluating column means for each ID, it creates a new column with called mean.

I tried adding a piece of code I have used earlier for calculating grouped column means in a function:

row_mean <- function(df, var, ...) {
summarise(
    across(
        .cols = everything(), 
        .fns = ~ mean(as.numeric(.x), 
            na.rm = TRUE)
    )
)}

distinct(a, ID, row_mean(a,ID), .keep_all = TRUE)

A data.frame: 3 × 3
ID  x   y
<dbl>   <dbl>   <dbl>
1   1   1
2   2   3
3   3   5

However, it only returns the first row of each ID value, as the default function does.

What I want:

A data.frame: 3 × 3
ID  x   y
<dbl>   <dbl>   <dbl>
1   1   1.5   # <-- column means for each unique ID
2   2   3.5   # <--
3   3   5.5   # <--

Any suggestions?