Ok, now don't go 'Oh well look at that, I am sub 16' on me.. LOL .
Well... LOL... I haven't done full 3x3 sessions for a while, but I have done some F2L sessions and I do believe my lookahead has been improved by bigger cube practice.
My average for pairs only has come down from mid-9s to high-8s, and generally feels more composed.
My theory is as follows. On big cubes, I do rely more on lookahead because I can't turn as fast. My pairs average on 4x4 is around 11 which is not that much slower than on 3x3, because although I turn slower, I pause less. Lookahead is also exercised in other steps like edges. All this does improve lookahead on 3x3.
However, after a few days on bigger cubes, I get used to turning less fast, and that does impact on 3x3 when I first go back to it. I try to start with a few TPS-spamming type things like timed PLL attacks to get back up to speed.
Finally, at our level in 3x3, I think the reason it seems a hard barrier is that we are at the point where any delay kills a solve - whether it's failed lookahead, slow recognition, lockups or fumbles, etc. We can no longer rely on just getting faster - we have to avoid things that lose us time too, even a fraction of a second. That's more difficult. On bigger cubes, that's less the case, as there's more time to recover from small delays. On bigger cubes I know I hesitate a little more to make sure I don't screw things up, and that habit harms 3x3.