Usually, distinguishing symptoms clients are worried about from the actual underlying problems requires an analysis, which is a real job requiring compensation. This will chase away some clients, but starting things up with the precedent of free work is not a great solution either.
It’s as close to an actual chicken-or-egg problem as it gets in many areas of business. Professionals often react out of fear and act like clients giving up after the initial analysis is their own risk, which is not a rule unless you act like it is. The client will usually play along and act like it’s your risk too if they smell fear, so it’s important to set the tone of cooperation right from the get-go.
In my own work, I have grappled with this issue many times and made many mistakes along the way, of course. These kinds of questions never have a perfectly clear and final solution, but I can tell you how I do it when my money is on the line:
These days I charge for analysis but count it toward the final cost if the project is realized. So if I charge 1500$ for understanding the problem and some quick wins, and the full project is 10000$, the client is only required to pay the 10k in total for the whole project, effectively making the analysis free of charge – but only if the full project actually takes place.
You will know what your industry will tolerate, but when in doubt – share the risk.