I think you have understood the topic in the wrong way. Basically, the across() and cur_data() are not at all related or interchangeable because they both are for different purposes in R programming.
The across() is used to implement a function on multiple columns whereas the cur_data() returns the current data for a group without the grouping variable.
For example,
library(dplyr)
mtcars %>% summarise(across())
The above code returns the same mtcars data set as it is because the default values in across are .cols = everything() and .fns = NULL. So it implements the NULL function to all the columns.