have arbitrary executable inherit errexit, if script is bash

Disclaimer: Never, ever do this! It's a huge disservice to everyone involved. You will introduce failures both in scripts with meticulous error handling, and in scripts without it.

Anyways, no one likes being told "don't do that" on StackOverflow, so my suggestion would be to identify scripts and invoke them with their shebang string plus -e:

for f in ./*
do
  # Determine if the script is a shell script
  if [[ $(file -i "$f") == *text/x-shellscript* ]]
  then
    # Read the first line
    read -r shebang < "$f"

    # The script shouldn't have been identified as a shell script without
    # a shebang, but check anyways
    if [[ $shebang != "#!"* ]]
    then
      echo "No idea what $f is" >&2
      continue
    fi
    # Strip off the #! and run it with -e and the file
    shebang=${shebang#??}
    $shebang -e "$f"
  else
    # It's some other kind of executable, just run it directly
    "$f"
  fi
done

Here's a script with correct error handling that now stops working:

#!/bin/bash
my-service start
ret=$?
if [ $ret -eq 127 ]
then
  # Use legacy invocation instead
  start-my-service
  ret=$?
fi
exit "$ret"

Here's a script without error handling that now stops working:

#!/bin/sh
err=$(grep "ERROR" file.log)
if [ -z "$err" ]
then
  echo "Run was successful"
  exit 0
else
  echo "Run failed: $err"
  exit 1
fi