Create a matrix with lowest cell count for every pair of binary variables

I have a dataset with several binary variables (x1-x5, values: 1, 2, NA). My goal is to identify whether pairs of binary variables have zero or very low cell counts in the cross-tab table (after ignoring the missing values). So, I would like to calculate the cross-tab table for each pair of binary variables in my data set, extract the lowest value from each table, and report the lowest value from each cross-table into a matrix. By doing so, I would have something similar to a correlation matrix where, instead of correlation coefficients, I would be able to look at the lowest cell count for each pair of variables. Below I created a toy dataset for anyone who will decide to help.

library(tidyverse)
    
df <- data.frame(x1 = rbinom(n = 1000, size = 1, prob = 0.5),
                     x2 = rbinom(n = 1000, size = 1, prob = 0.3),
                     x3 = rbinom(n = 1000, size = 1, prob = 0.4),
                     x4 = rbinom(n = 1000, size = 1, prob = 0.2),
                     x5 = rbinom(n = 1000, size = 1, prob = 0.05)) |> 
      mutate(across(everything(), ~as.factor(.))) |> 
      mutate(across(everything(), ~recode(., "1" = "2", "0" = "1")))
    
df1 <- as.data.frame(lapply(df, function(cc) cc[ sample(c(TRUE, NA), prob = c(0.85, 0.15), size = length(cc), replace = TRUE) ]))

I think this is what you mean. It's inefficient (we should only compute one triangle) but short.

cfun <- function(i, j) {
   min(table(df[[i]], df[[j]]))
}
outer(1:ncol(df), 1:ncol(df), Vectorize(cfun))

If you want to be more efficient:

n <- ncol(df)
m <- matrix(NA_integer_, n, n, dimnames = list(names(df), names(df)))
for (i in 1:(n-1)) {
   for (j in (i+1):n) {
      m[j,i] <- cfun(i,j)
   }
}