Honestly, I would not worry so much about strategies but instead apply to a major in which you are most interested. Your application will be stronger if it’s focused on your interests and you can explain how the particular schools you are applying to fit into those interests. Having EC’s in other areas in high school is fine, it shows you were exploring other interests. It shows you’re well rounded. Lots of engineering kids were in sports or music or whatever in high school and that doesn’t hurt their applications. I’m not sure why the opposite woudn’t be true.
While I agree that it’s generally a lot harder to move from LAS to engineering than the other way around, it’s also often a lot harder to get admitted to an engineering program. Also, if you enter as a freshman engineering major to a T20 school your coursework will be pretty darn set for the first year or so with STEM classes and you might have little chance to take classes in LAS subjects until later. My D is a freshman engineering student at a T20 and her freshman year is all Chem, Physics, Math, etc. I think the only non-STEM classes she’s taking this year is a required writing class and a 1 credit fitness class she wanted to take.
I agree with other posts though that you might want to see if there are ways to combine your interests or have a major/minor in different subjects. At many schools students combine varied interests in this way. I remember at the Northwestern info session they talked a lot about students having the flexibility to do this and provided some interesting examples of the different paths students took. Other schools, like Rice, don’t really admit to a major or particular college (except for music and Architecture I believe) . While they want to hear how you will pursue your interest once there and how your interests fit into what they offer, they don’t require you to declare a major until the end of your sophomore year.
So, my advice is to think about what your interests are and start doing your research on what different schools offer. I’d be less worried about a T20 than finding a good fit.