I've seen different responses from many different people on the subject of strategic thinking and its correlation to functional strength and preference in socionics. Some argue it's the LII who is most strategic whereas others argue it is ILI. Personally, I think that both are strong and correct cases, implying that both have the potential to be strong strategic thinkers, but I want to present my own interpretation of the functions and how they manifest in this subject area.
Both ILI and LII have 4D Ni and Ti with neither of the types ignoring either one of those functions. Thus, both types would be strong at determining patterns of the past and extrapolating them into the future as well as deterministic analysis from a set of initial principles. The former helps for strategy due to its ability to make accurate forecasts, whereas the latter can calculate in the short-term very precisely, making assumptions that are needed. Both combined would thus give a strong handle on what will be likely to occur in the near and further future.
However, whereas ILI uses Te alongside these two, LII uses Ne. This gives the ILI the advantage in devising pragmatic methods for achieving profit it sees using Ni and Ti and the LII the advantage in seeing more potential outcomes, which gives them the stronger ability to identify and seize opportunities.
Personally, I don't buy the idea that Ni types lack "big picture" thinking. If Ni is about perceiving the flow of events over time, then what part of that isn't able to see the big picture - the detail-orientedness? It's not as if Ni can't detach from the details. It is, after all, an intuitive function, meaning that it makes guesses, picks up on patterns, and leaps from one point to another. It's just that the ILI would be critical of those jumps until all the details are worked out. That isn't the same as not being able to see the bigger picture. "Going off your gut" isn't high intuition since any reasonable person would realize that saying "my gut feeling was wrong" is not a justification of why your plan failed. In other words, intuition is about having those "gut feelings/perceptions" rather than always acting on them. If you're doing something important, you have to think it through, and you better not make a mistake.
In this sense, I think that ILIs have the advantage since their plans are more likely to be better thought through.
However, holographic panoramic thinking is excellent at seeing the same circumstance under a new light. Thus, whereas an ILI would construct a more linear plan with steps A,B,C, etc, an LIIs thinking would be more nonlinear and creative, which would allow for innovative (but not necessarily pragmatic) methods. For example, in mathematics, whereas an ILI would start from the definitions and work up to a conclusion in a linear manner, the LII would observe a symmetry or interesting feature of the system they are working with and exploit it for full benefit, often leading to a more efficient (but not necessarily more effective) solution. It's less reliable but certainly useful in circumstances that call for novel ideas.
Personally, I think that this style of thinking is more useful in academia than in most strategic endeavours since there is usually a very large risk associated with completely novel methods of doing something in strategy. But, when done properly, it could be a big win.
According to Gulenko, ILIs are the more likely strategist of the two types, as explained in this reddit post https://www.reddit.com/r/JungianTypo..._of_vaserlans/ which I find to be quite accurate in my experience.
LIIs are more orderly and conscientious due to being a rational type but not necessarily more strategic. ILIs tend to detach and forecast more, but that doesn't necessarily make them planned. I think that both types have the necessary strengths to be strong strategic thinkers, but, like Gulenko says, I think it's more likely that an ILI would actually become one.
TL;DR: Whereas the LII has the strength of novelty and generality on its side, the ILI has the strength of accuracy and pragmatism on its. While both types have a strong capability to be strategists, in my opinion, due to LII's static nature and unvaluing of Ni and Te, the ILI has the advantage.
EDIT: A number of my thoughts on this subject have changed since the posting of this thread (including what I've written in responses on this thread). If my future posts reflect that, then this edit is to cover my ass.