I'm in the exact same boat OP! Exact to a carbon copy same boat!!
I just got my Exec to 6 stars and like you I can't beat 4 and 5 star Execs. There's a 4 star I could beat every time as 5 star, now hording MY extra 50 crystals a day!
I really can't get my head around the strat differences here. As a 5 star Exec went first against the 4 star (just like my 6 star now), but I can't win where I could with little difficulty.
Maybe it's starting with the Xanadu Blood with a 6 star that's a bad idea. As that's when it seems to go wrong.