Another vision, using which to work with the logical class of the matrix:
colnames(m)[aggregate(col~row, data=which(m, arr.ind = TRUE), FUN=min)$col]
#[1] "A" "B" "C" "B" "B" "A"
We get the indices of the TRUE values and then find the minimum (index of) column in which they occur, by row.
benchmark
library(microbenchmark)
n <- matrix(FALSE, nrow=1000, ncol=500) # couldn't afford a bigger one...
n <- t(apply(n, 1, function(rg) {rg[sample(1:500, 1, replace=TRUE)] <- TRUE ; rg}))
colnames(n) <- paste0("name", 1:500)
akrun <- function(n){colnames(n)[max.col(n, "first")]}
cath <- function(n){colnames(n)[aggregate(col~row, data=which(n, arr.ind = TRUE), FUN=min)$col]}
all(akrun(n)==cath(n))
#[1] TRUE
microbenchmark(akrun(n), cath(n))
# expr min lq mean median uq max neval cld
#akrun(n) 6.985716 7.233116 8.231404 7.525513 8.842927 31.23469 100 a
# cath(n) 18.416079 18.811473 19.586298 19.272398 20.262169 22.42786 100 b
