There is one defense of the modern usage of the word that my housemate and award-winning debater put forth.
“OK,” he said, “But ‘problem’ has a certain amount of intimidation value. If I can toss ‘the Problem of Evil’ at you, that has more force than, ‘the Question of Evil’, or worse yet, ‘How is it that a good god can exist when there is such heinous suffering in the world?’ The latter actually offers you the opportunity to respond.”
My response is to agree that “problem” is a bit more intimidating. But, learning is not competition. Despite the existence of the “sport” of debate, the appropriate use of argument, the activity of persuading and being persuaded, is to know the truth and teach it.
I think people find what they’re looking for. If I, with my friends (or enemies), am not looking to know the truth, I will not find it. If we are looking to find the truth, we will. (It may take awhile). If we are fighting over who has the biggest problem to deal with, laughing at our opponents’ consternation, perhaps we will neglect to solve our own.