But the exposure must be controlled, otherwise the person suffering risks making the problem worse.

Consider the example of Bob, a hypothetical person who has a phobia of elevators.

People tell Bob to face his fears. So, Bob decides to do this. He visits a local mall that has a bank of elevators. As he approaches them his body starts releasing adrenaline. His palms get a bit sweaty. His heart starts pumping a bit faster. His breathing changes. His anxiety levels raise. Bob tells himself to just keep going. He reaches the elevators, and reaches out a shaky hand, and pushes the button. He has thoughts - "I'm going to get trapped in this elevator" (for example), but he pushes those thoughts aside. The elevator arrives, with the announcement "Ding!", the doors slide open, and Bob gets in. Bob's thought's are crowding his mind, but he thinks "This is stupid, my fears are irrational" - but that does nothing to stop his fear. He pushes a button for a floor a few levels higher. The doors slide close. The elevator starts with a small lurch. Bob's anxiety levels are now really high. He is near panic. The elevator reaches his selected floor, the doors open, and Bob rushes out. As he leaves the source of danger his body floods with endorphins, and the further away he gets from the elevator the better he feels. His breathing calms down; his thoughts subside; his heart rate drops back to normal.

Bob has not conquered his fear of elevators. Bob has made his fear worse.

What Bob should do -- and these might seem like minor changes -- is to add some time for him to consider his thoughts.

Bob should spend a few moments looking at the elevator. Bob is in control - he can get in or not, as he wishes. He has to think about what his emotion is ('Fear') and what the thought causing that emotion is ('I will be trapped in the elevator'). He should sit with that for a few moments. Bob is not supposed to think "my fears are irrational". People do after all occasionally get trapped in elevators. (And spiders can be dangerous, and dogs do bite people, and sometimes blood does carry diseases, etc etc.) He should consider the evidence for that fear - ('I read that report of someone who got trapped in an elevator for a few days'). He should also think about how his body feels, and recognise that his heart rate has risen, and his breathing is different. Then he needs to think about different evidence ("I don't know anyone who has been trapped in an elevator. I have never heard of these elevators not working." and see whether that has reduced his fear a bit. This is an iterative process, you're supposed to use it each time you face that situation and it gets easier to do over time. Once Bob gets into the elevator he is supposed to stay in it until he has controlled his breathing and reduced his fear a bit.

So, yes, it's important to not avoid all sources of trauma but it's important to do it properly.

Trigger warnings do serve a useful purpose when talking about suicide or self-injury, which both have elements of contagion.

EDIT: Your article starts with an example about OCD. I agree that trigger warnings don't help there, and what people actually need is much easier better quicker access to treatment. There is very little risk of someone with OCD doing the things they have obsessive thoughts of (women have thoughts of putting their babies in microwave ovens, for example, but they would never actually do this). The article's expert then only considers trigger warnings for trauma - (where I agree they serve little purpose) but does not consider trigger warnings for contagion of self-harm or suicidal ideation.