This sounds really patronizing. I would be careful with that phrasing. People who don’t want their problem solved often know the solutions and don’t like the tradeoffs or change they entail. It is often not a knowledge problem.
In this way, listening and solving aren't too dissimilar. Simply listening can give the speaker an appropriate environment in which to solve their problem. A listener can play a part in helping to solve the problem, but helping foster the environment in which the problem can be solved. Don't mistake this as a silver bullet, but simply recognize that being a listener is an underappreciated role and listener vs solver isn't as dichotomous as it sounds.
Not really. The question, more generally, is: what should my role be in this conversation? Should I be an active participant in solving the problem? Or should I support you as you work through it?
As with so many things in life, the hard part is working out if your actions should be motivated by actually helping or feeling like you helped.
I think generally the person you’d say this to is already aware how the conversation will go if you don’t clarify beforehand.