Direct transmission through the earth could be as much as a quarter second faster than around the circumference. A hypersonic missile can travel about a half of a kilometer in a quarter second. So, any application where a half kilometer or quarter second head start might make a difference. The only thing I can think of where the stakes are high enough and timing critical enough is nuclear deterrence. But as hypersonic missiles and planes get faster, there could be other advantages.
LEO orbit period is 90min, so it takes at least 40min for a missile to go halfway around the globe.
Going in straight line across the earth at the speed of light takes 42ms. Going around the earth at the speed of light takes 66ms. Let's be extremely pessimistic and double that estimate at 132ms. Going through the earth saved you 90ms in this best case.
You saved 90ms over 40min. A ratio of 0.0000375, or 0.00375%.
There is no way it makes any sense to spend billions to save 0.00375% in response time to an incoming missile.
Your analysis is valid as long as one of the following are true:
1) We learn to bend light around the surface of the earth (otherwise relays are needed, which add significant delay).
2) hypersonic missiles are required to a) go into LEO and b) go all the way around the earth to benefit from a quarter second head start
I still think military warning applications are much more likely than someone trying to do HFT on the NYSE from Australia. But, like I said from the beginning. Neither are feasible because they need a super collider and a massive tank detector.
