Thanks @John Polo for spotting an issue. Eventually, I realized there were other two important problems in the loop I was creating:

1. The subset() call was not triggering any warning so it could not work properly anyway. In order to trigger it, I had to call values() but to avoid printing the results I embedded it into invisible(). I could have called other functions too to trigger the warning, but I opted for values() believing it to be pretty fast.

2. You cannot assign a matrix to a SpatRaster layer. So what I did was just to subset the original SpatRaster to that specific layer and fill it with NAs.

Here it is the solution to my problem in the form of a function. Hopefully others might find it helpful, even though I am sure some spatial analysis experts could come up with a way more efficient version.

# Function to repair a possibly corrupted SpatRaster
fixspatrast <- function(exporig) {
  
  # Create a new SpatRaster with same structure, all NA
  expfix <- rast(exporig, vals = NA)  
  
  for (lyr in 1:nlyr(expfix)) {
    tryCatch({
      lyrdata <- subset(exporig, lyr)
      invisible(values(lyrdata))   # force read (triggers warning/error if unreadable)
      expfix[[lyr]] <- lyrdata
    }, warning = function(w) {
      message("Warning in layer ", lyr, ": ", w$message)
      nalyr <- subset(exporig, lyr)
      nalyr[] <- NA
      names(nalyr) <- names(exporig)[lyr]  # keep name
      expfix[[lyr]] <- nalyr
    })
  }
  
  return(expfix)
}