Pivoting wide to long format and then nesting columns

A tidyverse approach to achieve your desired result may look like so:

library(tibble)

df_1 <-
  tribble(~key, ~values.male, ~values.female, ~values.red, ~values.green, ~value,
          "gender", 0.5, 0.5, NA, NA, NA,
          "age", NA, NA, NA, NA, "50",
          "color", NA, NA, TRUE, FALSE, NA,
          "time_of_day", NA, NA, NA, NA, "noon")

library(tidyr)
library(dplyr)
library(purrr)

df_pivoted <- df_1 %>% 
  mutate(across(everything(), as.character)) %>% 
  pivot_longer(-key, names_to = "level", names_prefix = "^values\\.", values_drop_na = TRUE) %>% 
  group_by(key) %>% 
  nest() %>% 
  mutate(data = map(data, ~ if (all(.x$level == "value")) deframe(.x) else .x))
df_pivoted
#> # A tibble: 4 x 2
#> # Groups:   key [4]
#>   key         data            
#>   <chr>       <list>          
#> 1 gender      <tibble [2 × 2]>
#> 2 age         <chr [1]>       
#> 3 color       <tibble [2 × 2]>
#> 4 time_of_day <chr [1]>

EDIT Following the clarification in your comments on the desired result we could simply get rid of the map statement as the end (which basically was meant for converting the tibbles for categories without levels to a vector) and add a mutate statement before nesting to replace the level with NA for categories without a level:

pivot_nest <- function(x) {
  mutate(x, across(everything(), as.character)) %>% 
    pivot_longer(-key, names_to = "level", names_prefix = "^values\\.", values_drop_na = TRUE) %>% 
    group_by(key) %>% 
    mutate(level = ifelse(all(level == "value"), NA_character_, level)) %>% 
    nest() 
}

df_pivoted <- df_1 %>% 
  pivot_nest()
df_pivoted
#> # A tibble: 4 x 2
#> # Groups:   key [4]
#>   key         data            
#>   <chr>       <list>          
#> 1 gender      <tibble [2 × 2]>
#> 2 age         <tibble [1 × 2]>
#> 3 color       <tibble [2 × 2]>
#> 4 time_of_day <tibble [1 × 2]>
df_pivoted$data
#> [[1]]
#> # A tibble: 2 x 2
#>   level value
#>   <chr> <chr>
#> 1 male  0.5  
#> 2 male  0.5  
#> 
#> [[2]]
#> # A tibble: 1 x 2
#>   level value
#>   <chr> <chr>
#> 1 <NA>  50   
#> 
#> [[3]]
#> # A tibble: 2 x 2
#>   level value
#>   <chr> <chr>
#> 1 red   TRUE 
#> 2 red   FALSE
#> 
#> [[4]]
#> # A tibble: 1 x 2
#>   level value
#>   <chr> <chr>
#> 1 <NA>  noon

df_2 <- tribble(~key, ~value, "age", "50", "income", "100000", "time_of_day", "noon")

df_pivoted2 <- df_2 %>% 
  pivot_nest()
df_pivoted2
#> # A tibble: 3 x 2
#> # Groups:   key [3]
#>   key         data            
#>   <chr>       <list>          
#> 1 age         <tibble [1 × 2]>
#> 2 income      <tibble [1 × 2]>
#> 3 time_of_day <tibble [1 × 2]>
df_pivoted2$data
#> [[1]]
#> # A tibble: 1 x 2
#>   level value
#>   <chr> <chr>
#> 1 <NA>  50   
#> 
#> [[2]]
#> # A tibble: 1 x 2
#>   level value 
#>   <chr> <chr> 
#> 1 <NA>  100000
#> 
#> [[3]]
#> # A tibble: 1 x 2
#>   level value
#>   <chr> <chr>
#> 1 <NA>  noon

One option that will return the same type of output as the supplied input:

df_1 %>%
 group_split(key) %>%
 map_dfr(~ select(., where(~ !all(is.na(.)))) %>%
          pivot_longer(-key, names_to = "level", names_prefix = "^values\\.") %>%
          summarise(key = first(key),
                    vals = if(n() == 1) list(value) else list(tibble(level, value))))

  key         vals            
  <chr>       <list>          
1 age         <chr [1]>       
2 color       <tibble [2 × 2]>
3 gender      <tibble [2 × 2]>
4 time_of_day <chr [1]>  

The structure of output:

$ key : chr [1:4] "age" "color" "gender" "time_of_day"
 $ vals:List of 4
  ..$ : chr "50"
  ..$ : tibble [2 × 2] (S3: tbl_df/tbl/data.frame)
  .. ..$ level: chr [1:2] "red" "green"
  .. ..$ value: logi [1:2] TRUE FALSE
  ..$ : tibble [2 × 2] (S3: tbl_df/tbl/data.frame)
  .. ..$ level: chr [1:2] "male" "female"
  .. ..$ value: num [1:2] 0.5 0.5
  ..$ : chr "noon"