If those cheap camera modules would incorporate the same technology they most probably would. As you know yourself they don't.
It's more complex than that. Consider the Nikon V1/J1. Relatively small sensors (crop factor of 2.7) and read what reviewers say about their AF performance:
"The J1 and V1's contrast-detection AF systems, which take over when light levels are too low for effective phase-detection AF (and remember that this is the camera's decision, not the photographer's) are less impressive. Focussing is less positive (AF wobbles quickly around the target rather than locking straight on it) but impressively accurate when it gets a 'lock'. The problem is that even in average interior room light, both cameras struggle to find that lock. They rarely give up completely (and built-in AF illuminators are on hand in both cameras) but we've been surprised by how frequently they falter."
A far cry from AF performance of Oly/Panasonic/Sony mirrorless cameras.