Relic Shield does not execute on Thresh because Thresh is a ranged champion in every single possible mechanic in the game. The game engine NEVER considers Thresh a melee champion. His attacks are in the same group as Kayle's Righteous Fury and Vel'Koz attacks, as they are ranged attacks that are not projectiles.
Your assumption that Thresh cannot buy Runaan's Hurricane is wrong: he can buy it and it works, firing two extra bolts every time Thresh attacks. There have been patches where Thresh couldn't purchase Runaan's Hurricane, but it has been corrected since patch 5.23 (section Bugfixes).
As every other ranged champion, Thresh can use Targon's line (Relic Shield and its upgrades) but do not receive the bonus execution on lane minions. It's a very common item on Thresh because of the desired extra health and the fact that his Flay passive will usually store enough damage to easily kill a minion without using mana.