Pick and choose. I choose both methods! There is plenty of time. Actually if you guess the wrong lead angle your gyro angle would be less than 5º off with a pretty terrible guess. That won't impact the accuracy of your shot.
We fuss and pick when we're making these tutorials, anticipating that equally fussy people will say, "You did this wrong! Your shoot bearing should have been 349º, not 350! You idiot!" Actually a torpedo with a 1º gyro angle is functionally as accurate as one with a zero gyro angle. So is one with a 5º or a 10º gyro angle. You'll NEVER be 10º off using a rule of thumb.
Rules of thumb allow you to be just as accurate as the guy with the vector analysis who sends his torpedo straight up the zero line, but with rules of thumb you don't have to remember how fast that Mark 10 is, or make the mistake of selecting slow speed for your Mark 14 and drawing the vector analysis for fast speed. You can verify the bearing of the torpedo track on the attack map.
In shooting, some things can be close enough, and others need to be accurate. In the John P Cromwell attack, speed measurement is more forgiving than the Dick O'Kane attack, but the AoB measurement is more critical. Fortunately, in both types of attack our AoB can be 100% accurate and calculated in your head in less than a second. There is no guesswork there, just a simple to remember rule.
In the Dick O'Kane attack you are attacking 90º from the target track. The AoB is 90º minus the difference of your shoot bearing from zero. A shot sighted up the 350º bearing line is ten degrees from zero (360). It has an AoB of 90º minus 10º equals 80º port or starboard, depending on which side of the target you are looking at.
In John P. Cromwell you are attacking 45º from the target track. The AoB is 45º minus the lead angle you select. A shot sighted up the 350º bearing line has an AoB of 45º minus the 10º lead angle equals AoB 35º starboard or port, depending on which side of the target you are looking at.
These rules are neither difficult to calculate without a calculator, nor difficult to remember. They are exact results, not approximations.
But both methods are prone to error. The vector analysis method is prone to the error of setting wrong torpedo speeds, or misremembering the torpedo speed. This error will result in a miss.
The TDC/rules of thumb method is prone to an error of getting the gyro angle a couple of degrees from zero. This results in a sunk target and no knowledge of the error at all unless you investigate the results later or check the attack screen before you shoot.
I'll take the error that sinks the target! Your results may differ.
And there's plenty of time to do both! The video shows you how to do the vector analysis right on your nav map without pencil and paper. Use both and you check the possible errors of both methods with the strength of the other. That's the policy I recommend. These are complimentary, not competitive methods.