What are the major ethical concerns with private tutoring?

You seem to have a pretty good ethical sense based on your question, so I think you won't go far wrong following your instincts.

But if you have the opportunity, it is a good idea to go visit the prof teaching the course for which you are tutoring and ask for any additional general guidance.

It is usually a good idea to answer "trivial" questions with other questions. "What do you think the answer is? Why? Explain your reasoning." That sort of thing. Often the student will get a flash of insight. But if you can point to a flaw in reasoning you help the student without giving direct answers.

Good professors when asked questions by students seldom give out complete answers unless it is in the context of a continuing conversation. Instead, a "minimal" hint is given to get the student over a block and which lets them proceed. If a student repeatedly has problems with the same issue or block, an additional exercise will be given that tries to give the student practice with the reasoning.

But the technique of asking for an analysis of any issue that leads to a block is a good way to diagnose the student's thinking and help them get over it.

Giving students general problem solving strategies is also good, but probably less useful, as it doesn't usually have a diagnostic element. But even "Where did you look for an answer?" can help. Especially if followed by "Where else could you look?". You could then suggest other sources if appropriate.

I'll note for completeness that some students are misled by the resources that they use. Some of those resources are wrong and some may be written poorly leading to ambiguity. I've had students get in terrible jams by using perfectly rational thought processes applied to bad input data. That is rare, but it can occur.

@Buffy has hit all the points. But I'll add that I've noticed that different departments have quirks. Often the quirk is just the department heads personal bias. It could be that at Nebraska College, tutoring students in your own department is fine, as long as there's no conflict of interest. But at Kansas College, the department head thinks that the same situation is unethical (I mean, after all, the undergrad has already paid tuition which, in part, pays your stipend, and now, here you are double-charging him.)

So my advice is that you ask the department head his views. His views may be unreasonable, but....you know.