So Greymule was correct that in both OBR and FED, by calling time, the offense gets the benefit of being able to return to the missed base without liability of being called out?

No. mbyron pointed out that by asking for time, the BR has announced that he has completed his advancement. So even if he returned to the bag, he could be declared out on appeal. The offense can't make a mistake and then ask for time in the hope that the umpire will grant it and thus allow them to correct their mistake with impunity.

Ok to summarize...

OBR: Safe
FED: Out

Correct?

Not quite.

For OBR I would not grant time immediately, since doing so gives the offense an advantage. Since the ball is still live, the defense can still appeal. If the defense gives no indication of wishing to appeal, I will grant the request for time.

For FED the BR is out if the appeal is properly made before BR returns to the base.

Thank you. I agree with you, with the caveat that the appeal is before the runner returns to first.