Why doesn't Relic Shield execute always work on Thresh?

This confusion comes from misinformation that Thresh is no longer able to build Runaan's Hurricane. This bug was fixed in patch 5.23.

The Relic Shield appears to execute properly at times solely because of the bonus damage from Flay.

The important thing to remember is that the Relic Shield effect still works even if you don't proc the execute. As long as you get the last hit, it will proc its heal/gold share. The last hit for the heal/gold proc can be performed with any source of damage whether from auto attack or skill.

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.