Probably not. We are talking about some function of first N numbers and its limit when N approaches infinity. There is more than one reasonable way to clarify "first prime factor" and "first N numbers", but I think for all of these reasonable definitions the median diverges, having different subsequential limits for even and odd values of N.
I'm taking the median of the limiting distribution, and I agree that the sequence of medians you describe does not converge.
How so?
1 has no first prime factor so must be excluded along with 0. 2 is always >=50% of the head sequences of the rest: