Magic the Gathering - What to do with permanents that only work "until end of turn"?
An enchantment is a permanent, and permanents remain on the battlefield indefinitely. Only another effect can move them to another zone.

110.4. There are five permanent types: artifact, creature, enchantment, land, and planeswalker.
110.1. A permanent is a card or token on the battlefield. A permanent remains on the battlefield indefinitely. A card or token becomes a permanent as it enters the battlefield and it stops being a permanent as it’s moved to another zone by an effect or rule.

You are correct that once a Trial has entered the battlefield and done its thing, it's practically useless. When you play a Cartouche and return the Trial to your hand, you can cast the Trial and get its effect again. You can play the Cartouche the same turn or any number of turns later. The Trial will remain on the battlefield until something happens to it.
The idea behind the &quot;Trials&quot; card cycle is that you get a one-shot effect that you can reuse with Cartouches, with the downside being that the combo can be broken by e.g. destroying the Trials while it's sitting uselessly on the battlefield.