Why does this mergesort variant not do Θ(n) comparisons on average?
Since you don't do any reordering while splitting, the length of array after the while loop can not be larger than the length of the longest increasing subsequence in the input. Since that one is on average about $2\sqrt{n}$ elements long¹, you keep too many elements in unsorted. 

In particular, there is no $b$ so that your ansatz describes the actual number of comparisions.

Assuming that the probability distribution of the length of the longest increasing subsequence survives your splitting, a proper ansatz would be of the form

$\qquad\displaystyle C(n) = C(n - 2\sqrt{n}) + \Theta(n)$

which does not seem to solve² to $C \in O(n)$.

In fact, it's even worse since you keep the "first" increasing subsequence, not the longest (in general); consider for instance $[1,n,2,3,4,\dots,n-1]$; you keep $[1,n]$ and recurse on $[2,3,\dots,n-1]$.




On the distribution of the length of the longest increasing subsequence of random permutations by J. Baik, P. Deift and K. Johansson (1999) [via Wikipedia]
I have not solved it explicitly, but plots suggest as much.