In fact, there is another mechanic that proves they absolutely COULD do system checks (including save-outs) at the end of turns: The irony is that it's another thing they've implemented incorrectly, to the detriment of players!
A barbarian's rage is supposed to end early if they haven't attacked a hostile creature or taken damage since the end of their last turn. This means that a Barbarian's rage doesn't end as long as they are taking damage each round, from any source and at any point in the round, or if they are making opportunity attacks, even if they aren't attacking with their action each of their turn.
Barbarian rages and rushes into a breach point, attacking and felling the enemy there, and ends their turn; at initiative 20, they take minor damage from the burning pitch and falling embers that come from being exposed in the open of this particular battlefield. Their turn comes again; they are still raging, but they have a friend to rescue; they wade out into enemy lines, pull the injured person back and use their action to stabilise them, in the safer under cover area, then stand back in the breach, preventing enemies from overwhelming the point. They end their turn, and then proceed to weather the attacks of four or five other foes while the rest of the part scrambles to regroup behind them - they survive this because their rage halves all the incoming damage they take from those attacks. Their turn comes again; the foes are endless, but they only need to hold on another few seconds, before the rogue will have the quest maguffin ready - they take the dodge action, hold the breach and call to the cleric that they could use some help. Next turn, the cleric has given them enough of a shot to survive the attacks - many of them missed thanks to the defensive stance, but the passive damage is still nasty. They disengage from the fight and fall back to the magic circle which then whisks the party and their quest maguffin away just in time.
Now... that's how it should work...
It goes a little different: Barbarian rages and rushes into a breach point, attacking and felling the enemy there, and ends their turn; at initiative 20, they take minor damage from the burning pitch and falling embers that come from being exposed in the open of this particular battlefield. Their turn comes again; they are still raging, but they have a friend to rescue; they wade out into enemy lines, pull the injured person back and use their action to stabilise them, in the safer under cover area, then stand back in the breach, preventing enemies from overwhelming the point. They end their turn: Their rage immediately ends. They weather the attacks of four or five other foes while the rest of the part scrambles to regroup behind them, they take a hideous pounding because their damage resistance is gone; they only survive using their barbarian hail mary to survive on one hp from a lethal blow. Their turn comes again; the foes are endless, but they only need to hold on another few seconds, before the rogue will have the quest maguffin ready - They Rage again - their last rage for the day - and take the dodge action, holding the breach and calling to the cleric that they could use some help. Again, their rage immediately ends; they go down hard, and enemies begin to make it through to the rest of the party; the cleric tries to get the Barb up, but the already badly injured rouge goes down instead; people are dying, others are unconscious, and the maguffin isn't ready; the party is overwhelmed before they can escape.
Currently, in game, if a barbarian ends their turn without having attack or taken damage on their turn, their rage ends immediately at the end of their turn. This is clearly the game making a check at the end of your turn, and ending an effect based on whether the conditional is met or not - everything required for an end of turn save-out effect. It also means that Barbarians cannot rely on taking damage outside of their turn, or opportunity attacks to keep their rage ticking over, and it means that they immediately lose their damage resistance if they end their turn without attacking.
