This thread explains how V-bit tool paths are calculated:
This assumes a perfectly sharp tip to the cutter. If the tip of your cutter is not perfectly sharp (it won’t be) then when you zero the Z height, you will actually be a little bit lower into the material than expected, and your cut widths will be wider, causing an effect like you have seen. The same thing happens if your material has a varying thickness and you zeroed at a thinner spot. This will be the case whether you use the paper method or a probe for zeroing.
You could try compensating by moving your Z zero up slightly so that the bit won’t cut as deeply. That is essentially what @JoeyMoose is doing with his method, but you could jog up even very slightly more and then zero if necessary.