Showing x-axis on each facet in ggplot using facet_wrap

I'm trying to create a plot for a publication that requires faceting. It has to show the same x axis on each facet and maintain even spacing of each y-axis metric.

facet_grid allows me to create the y metric spacing I want, but I can't get the x-axis to show across each facet. This seems to be part of how facet_grid works though as per this answer

facet_wrap allows me to show the x-axis on each facet, but it disrupts the y metric spacing. The values of each x-axis are also different which I need to be the same.

facet_rep_grid seems like it should be a solution but it throws me an error "Error in axisgrob$height$arg1 : $ operator is invalid for atomic vectors"

Using patchwork seems like it could be a solution but this also disrupts the y axis spacing that I need to be the exact same for each grid.

Reproducible examples of the plots I just described:

install.packages("lemon")
library(lemon)
data("mtcars")

df <- mtcars %>%
  rownames_to_column(var = "car")

mt <- ggplot(df, aes(mpg, car, colour = factor(cyl))) +
  theme(strip.placement = "outside")+
  geom_point()

# Good y metric spacing, can't get x-axis to show on each
mt + 
  facet_grid(vars(cyl), switch = "y", scales = "free", space = "free") 

# Incorrect y metric spacing, shows x-axis on each but uses different scales
mt + 
  facet_wrap(~cyl,  scales = "free", ncol=1, switch = "y") 

# Throws an error

mt +
  facet_rep_grid(vars(cyl), scales = "free"
                 , switch = "both", space = "free")

facet_grid version:

facet_wrap version:

Your final solution works for me.

mt +
  facet_rep_grid(vars(cyl), scales = "free"
                 , switch = "both", space = "free",
                 repeat.tick.labels = T)

If you still are getting an error can you post it? I had to re-install rlang and vctrs packages to get it to work.