Conditionally fill a column downwards based on the presence/absence of information in another column

I am trying to fill a large file containing taxonomic information so that each row contains relevant data. At present, the file is structured, thus, from infraclass through to subspecies:

Here, for example, I would like to fill Genus downwards when there are data in Species, likewise for Family, and then remove the redundant rows. The end result should be something like this:

I've included example data, below:

df <- data.frame(
family = c("Rheaidae",NA,NA,NA,NA,NA,NA,NA,NA,NA,NA,NA),
genus = c(NA,"Rhea",NA,NA,NA,NA,NA,NA,NA,NA,NA,NA),
species = c(NA,NA,"americana",NA,NA,NA,NA,NA,"pennata",NA,NA,NA),
subspecies = c(NA,NA,NA,"americana","intermedia","nobilis","araneipes","albescens",NA,"garleppi","tarapacensis","pennata")
)

I've tried several different ways of combining mutate, ifelse, loops, etc. but I'm not getting anywhere. I've provided one example, below, which didn't work but was my most successful attempt, thus far.

df %>%
  mutate(g = ifelse(is.na(subspecies), "NA", 
            zoo::na.locf(species, na.rm = F)))

EDIT
@benson23 suggested using

df %>% fill(everything(), .direction = "downup") %>% distinct_all()

I'd come across something like this previously, but even though it does a good job of filling cells, it leaves me with erroneous rows. I've given an example, below:

Becomes:

Rather than:

In short, the data run-on and there's no clear way to remove the erroneous rows.

df %>%
   fill(family)%>%
   group_by(family)%>%
   fill(common:species, .direction = 'downup')%>%
   group_by(across(family:species))%>%
   reframe(across(everything(), ~if(all(is.na(.x)))NA else na.omit(.x)))

# A tibble: 4 × 5
  family        common    genus    species       subspecies
  <chr>         <chr>     <chr>    <chr>         <chr>     
1 Rheaidae      Rheas     Rhea     americana     americana 
2 Rheaidae      Rheas     Rhea     americana     intermedia
3 Struthionidae Ostriches Struthio camelus       australis 
4 Struthionidae Ostriches Struthio molybdophanes NA