Let me add a question to this. when using these charts and tools, they don't give you a minimum day. So the set and wait.....for how long, would be good to have a decent minimum. I understand its to taste but talking about carb'd and ready regardless if green or aged.
I think with "set and forget" (assuming your keg is cold - it often takes 12-24 hours or so just to cool it down from 70F to sub-40F), I would say 5-7 days to carb your beer properly.
With all the "my keg won't carb" questions, it's all about:
Pressure
Temperature
Time.
If you pressurize with CO2 at the right pressure, for a given temperature, and wait for some time - the keg will have to be carbed at desired level. If it is not, either pressure is not getting into the keg (or leaking out of the keg), or temperature is off, or you didn't give it enough time. But you should see big progress even over a day or two.
I use the system where instead of going for say 10 psi for 5-7 days at 36F, which is about 2.5 volumes of CO2, I would go for higher pressure - 30-35 psi or so, for first 24-36, maybe 48 hours if I am really impatient, and then back down to 10 psi. This way the beer is carbed within maybe 2 days or so (my beer is already cold and clear during transfer to keg). I almost never overcarb it using this procedure. It gets a little better at 5-7 days mark (and continues to get better), but it's very drinkable within a few days, as it basically carbed.
I don't like shaking the keg for numerous reasons: 1. it's strenuous (on my back) 2. it's not very controllable. 3. You are mixing in yeast, spent hops and other sediment into the solution 4. You are using up proteins that are needed for head retention. 5. You can end up with easily over-carbed keg, at which point you can relatively quickly de-carb it, by agitating it once again and releasing CO2, but it cost you more in head retention.
With IPAs and other beers (like Hefeweizens) where you want to drink them as fresh as possible, waiting even 7 days (never mind 21 days) is borderline sacrilegious.