Rakan's attacks (even up close) don't register as melee, therefore they don't execute minions. This is especially weird because he has a different animation and even does 2 more damage when he pimp slaps a minion.
Champions are always either melee or ranged (a few have abilities that swap them between this); getting closer, even if they have a different attack animation for it, does not change that. Rakan, while he does have a ridiculously short basic attack range, is considered a Ranged champion (for instance, he can utilize Runaan's but not Hydra).
Also note that even if you are Ranged you can still use the Relic Shield items, you simply will not get the bonus Execute damage. Last hitting minions any other way still grants the heal/shield and the FotM passive shield CDR.
He. is. not. melee.
Rakan is a ranged champion. You can't buy Titanic/Ravenous Hydra as a proof. Even with the melee animation, he isn't. Same for thresh.
