How to properly address an unsolved problem that is very hard to solve in mathematical thesis?

I am assuming you're talking about a presentation to be given at the defence of a PhD thesis.

Let me address a few points you raise.

• It is perfectly appropriate to include subjective judgements about the difficulty or significance of certain results in a mathematical thesis (or paper). Mathematicians make these judgements all the time in deciding what to work on and in evaluating other people's work. What matters is that you have a solid rationale for the judgements you make. In your situation, the rationale for considering the general case "very hard" is that it involves non-isolated singularities. That seems perfectly reasonable to me.

• If you're worried about presenting subjective judgements as if they were statements of fact, there are various standard forms of wording you could try. Instead of writing "solving this problem... is very hard" you could write "solving this problem... appears to be very hard" or "solving this problem...is generally considered to be very hard" (assuming the latter is true). If there are published works affirming that the general case is hard, you can cite them in support of your point.

• Finally, asserting that the general case of a problem is very hard does not imply (in any sense of that word) that a special case is easy, only that it is easier. But that is completely fine: experienced mathematicians are well used to people restricting themselves to more tractable special cases. I don't have any examples to hand, but I have read many papers in which the author does exactly this. This is not interpreted as an "admission" that the special case is "too easy". (In fact, often when someone does prove something that really is too easy, their rhetoric is the opposite: they try to puff it up to make it sound more difficult than it really is.)

I don't think you have anything to worry about. Presumably you and your advisor have agreed that the problem you did solve is hard enough to justify a thesis, else you would not be at the presentation stage. That's what your audience will judge. A small improvement in understanding the zeroes of the zeta function could make a fine thesis; no need to apologize for not settling the Riemann Hypothesis. Just put your work in context. Keep the audience interested in what you have done.

Finally, almost all the time the formal thesis defense is just a formality. You want to do it well, but need not fear failure.

Can you say something along the lines of "The general case is a long-standing open problem..."