You know the saying "culture eats strategy for breakfast" by Peter Drucker. It's so worn out, isn't it? Only because it's so valid. I'm sure you have seen it repetitively in practice. Strategic initiatives fail over and over again for the same reason. New seeds are planted in the same infertile soil, "we've tried that before and it didn't work". Are we surprised that the results aren't any different than before? The strategy fulfilling culture just isn't there.
Does that make the strategies invalid? It depends, but probably not. Making sound strategies isn't that hard, it's more about common sense once you have gathered the pieces and done the analyses. Making excellent strategies may be a bit harder, while you won't know how excellent they were until you see the results some years down the road - if you were successful in execution, your assumptions held and you didn't run into major risks.
Good execution isn't a one man show, and shouldn't require any superhero deeds either. It does however, require excellent cultural understanding at the helm of the ship. I say excellent because it's still so rare. What kind of culture supports our chosen strategy the best, and how far are we from that today? How long will it take us to transform, or is it better to revise the strategy? There isn't one super culture that fits all organizations and strategies, so it's not worth trying to copy the cultures of the famous few. You need to figure out the best recipe for your particular context.
So which comes first, strategy or culture? Does it matter? What matters is excellent alignment between the two. Start now. It's not a day too early.