`wait`

>and error out if at least one errors.

`wait` propagates the exit status of the waited task. `set -e` triggers the script to exit on any command failure.

>And try separating their output so that they don't get interleaved into a big mess.

`| while read -r line; do echo "task X: $line"; done`

Learn the tools instead of being so confident that it can't be done or that it's complicated.