Anonymous function not picking up arguments using mutate + map + mutate

I have to be missing something obvious here... I'm trying to add a column to each element of a list-column of data, where the new column name corresponds to the name of an element in another column.

Example data

library(dplyr, warn.conflicts = FALSE)
library(purrr)

df1 <- tibble(
    new_cols = c("y", "z"),
    data = list(
        tibble(a = 1),
        tibble(b = 1)
    )
)

df1

# A tibble: 2 x 2
  new_cols data            
  <chr>    <list>          
1 y        <tibble [1 x 1]>
2 z        <tibble [1 x 1]>

df1$data[[1]]

# A tibble: 1 x 1
      a
  <dbl>
1     1

Calling an anonymous function

So, I would like to add a column called "y" to the first tibble in the list, and a column called "z" to the second tibble in the list. I would like to accomplish this by calling an anonymous function, but the following results in an error:

df1 %>%
    mutate(data2 = map2(
        .x = data,
        .y = new_col,
        .f = function(x, y) mutate(x, {{ y }} := 2)
    ))
#> Error in (function (x) : object 'y' not found

map seems to be looking for an object "literally" called "y", which of course is not correct.

Using a function saved to an object

However, if I create a function and assign it to an object then I can map it to the data without any issues:

my_mutate <- function(.data, .colname) {
    
    mutate(.data, {{ .colname }} := 2)
    
}

df2 <-
    df1 %>%
    mutate(data2 = map2(
        .x = data,
        .y = new_cols,
        .f = my_mutate
    ))

pull(df2, data2)

[[1]]
# A tibble: 1 x 2
      a     y
  <dbl> <dbl>
1     1     2

[[2]]
# A tibble: 1 x 2
      b     z
  <dbl> <dbl>
1     1     2

Can anyone explain what I'm missing here?

Thanks!