There is to my knowledge no such utility, but it is easily implemented in a shell script.
A short shell script that implements what you described:
serve ./public & serve_pid=$!
You may want to insert a
sleep 3 call (or similar) after starting
serve in the background, to allow it to initialize properly before running the testing script.
$! will be the PID of the most recently started background job (the
serve process). When the
run-chrome-tests.sh script finishes, the script above will explicitly terminate the
serve process by signalling using