Re: "You take the 3 points and move on"
I think if you're playing Cincinnati & you're moping the floor with them, if you don't get a TD with the three downs you're allotted, you kick the field goal. Makes no sense to "go for it" on 4th down. You're not proving anything other than you couldn't do it in 3.
If you're playing the Patriots & it's 0-0, you take the 3. If you're down by 3, you take the three. If you're up by three, you kick the field goal.
All the while, you are trying to get a TD. but if you don't get it, you kick the field goal, unless you're down by 7 or more & running out of time.