SFI145, you're absolutely right.
But to calculate the said tailwind component (or headwind, for that matter, which would be a negative value from a TWC perspective) you need to take the cosine of the actual wind and the angle relative to your flightpath.
So the tailwind component of a 120/12 when you're heading (not bearing!) exactly 360 would be 12*COS(180-120), and equals 6 knots.
The EFIS indeed does this by calculating the difference between the TAS value from the ADC and the GS from the IRS/GPS. Looking at the difference between the plane's heading and bearing you can calculate the wind vector by reversing the trig function I posted earlier.
