Robust implementation of horizontal y-axis title above y-axis in ggplot

Question

I am attempting to develop a robust ggplot theme (or sourcable file) to override the default ggplot theme, with the new theme having the y-axis title horizontal above the y-axis text.

Related to Put y axis title in top left corner of graph, although I'm trying to overwrite the default ggplot theme and extend to a more robust and general solution.

Not a duplicate of Move Y axis Title horizontally ggplot2 as I'm looking for a automatic, general solution instead of trying to manually find numbers for every single plot.

The current implementation below has three issues:

Using theme_set to override the default doesn't trigger ggplot_add.test_theme to switch the y-axis and subtitle. I'm guessing theme_set acts as a fallback, so the default theme is never added to the ggplot, but I can't work out how to hook into the default ggplot build to make this change.
Creating a facet plot moves the y-axis outside the facet. This may be unfixable with the current approach of moving the y-axis into the subtitle. I have also tried adding an invisible secondary x-axis to add white space between the facet title and the panel/plot (with strip.placement = "outside"), and trying to add an annotation/tag into the gap, but I can't see a general way of finding the appropriate coordinates with grid/gtable
Switching the y-axis and subtitle in this way means the theme has to be added after any changes to the y-axis title through labs (or otherwise).

I'm open to completely different approaches.

library(ggplot2)

test_theme <- \(){
  tmp <- theme_grey() + theme(
    plot.title.position = "plot",
    axis.title.x = element_text(hjust = 1)
  )
  class(tmp) <- c("test_theme", c("theme", "gg"))
  return(tmp)
}

# Make changes to the plot when test_theme is added to a ggplot (e.g. p + test_theme())
ggplot_add.test_theme <- function(object, plot, object_name) {
  # Move the y-axis to the subtitle
  plot$labels$subtitle <- plot$labels$y
  plot$labels$y <- ""
  plot
}

# Set the new theme as the default
theme_set(test_theme())

# Problem 1
# x-axis moved, y-axis in default position
ggplot(mpg, aes(cyl, displ)) + 
  geom_point()

# both axis have moved - this is what the plot should look like.
ggplot(mpg, aes(cyl, displ)) + 
  geom_point() +
  test_theme()

# Problem 2
# Facetting moves the y-axis outside the facet.
ggplot(mpg, aes(cyl, displ)) + 
  geom_point() +
  facet_wrap(fl ~ .) +
  test_theme()

# Problem 3
# Breaks if labs is after the theme
ggplot(mpg, aes(cyl, displ)) + 
  geom_point() +
  test_theme() +
  labs(y = "Y test")

Robust implementation of horizontal y-axis title above y-axis in ggplot

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