Isn't that what tee is for? Like

$ prog | tee /tmp/log.txt

When I submit a process to bg, I mostly use cron and I do not care about seeing output during runtime.

So, tail suites my needs in the rare cases something unexpected seems to be happening.

That's if you start the process with advance knowledge that you'll want to tail the output and log it. Not if you want to view the output of an existing process