Ahhh....Grasshopper....now we understand where your coming from.
The example used by the YMCA instructor was probably intended to make the point of "Pay attention to your time and check against your planned time (if you're diving off of tables) frequently". And IMO, the bit about having to get assistance and additional breathing gas to get out of the predicament could have been left out (aka, not a detail appropriate for brand new divers).
I don't have my PADI tables handy, but if I remember correctly, 40 feet still gives you lots of time underwater. And scootching down to 44 is not the end of the world. Yes the plan had a hard bottom of 40 feet, but being at 44 for a small increment of time compared to the total dive time is not that big of an increase in exposure.
But your example there is also a wonderful example of how a dive computer will benefit you. As you already know, when you are on tables, you plan for a complete square profile. But when we are in the water, we are never completely square. It's the computer that will take over the monitoring of depth and time and present you with it's calculation of how much No Deco Limit (NDL) time you have left.
The big penalty of busting a plan is over staying your planned amount of time, both from a N2 absorbtion view point as well as remaining air.
Hypothetical #1: I'm at 70 fsw and engrossed in taking photos of fish that I don't realize that the current has moved me away from the anchor line. A check of my computer shows that I still have 10 minutes of NDL, but my SPG shows I'm marginal on air to make it back to the line and make a slow safe ascent, and I still have to hump through the current to get to the line.
What did I do wrong? I wasn't paying attention to the limiting factor which was air management. NDL was okay, but the SPG was the faster of my "clocks".
Hypothetical #2: I leave the camera on the boat and return to the bottom. This time I stick close to the anchor line and hardly exert myself (air consumption rate way lower than #1). This time the NDL is the governing clock and I need to begin moving shallower before it runs to zero. Yes, my air would allow me to stay longer, but that means busting the NDL's and that is a big no-no of the dive plan.
The deeper you go, the shorter your allowed time, and the more frequently you need check your computer's displayed NDL.
Besides, you're gonna be looking to check your depth anyhow (hey! I'm at 84 fsw! Cool!) and all the computers you are asking about display the current detph and NDL on the same display. So as long as you make sure that NDL clock never reaches zero, you'll be okay.
As far as packing tables, well, you can bring them along. But if you're under no deco restriction and have a good computer, you won't need them.
You got some good questions, but you might be getting ahead of yourself. (Kinda sounds like me not all that long ago

) Talk to your instructor with these questions. They love it when their o/w students come back and want to learn more about diving.