Because in science we tend to use "Occam's razor" approach - we don't make assumptions that are more than necessary to explain the behaviour of something. Most adherents of the anthropic principle assumes "Oh hey, we probably live in a multiverse" to justify it, but we have no strong evidence that that is true. So making that assumption is difficult to justify. It may be true! But we don't know it is, and until it can be proved, the theory rests on shaky ground. It might be that another, simpler theory can explain the same behaviour. Should we assume that a simpler theory is true? No, not necessarily. But it lies on the adherents of a more complex theory to come up with reasons why that theory is right, by making predictions that can be tested.

Because it's too powerful and can be used to explain almost anything. A good theory should make specific and rigid predictions with few free parameters.

As an analogy, I think everyone here would be quite upset if someone made the claim, "why shouldn't we trust Musk, Gates, Jobs, etc at their word when they say what caused them success? Obviously it is true because they are successful"!

If you reject the premise I proposed, but not the anthropic principle, then you have just moved the goal posts on when to accept blind tautologies without further examination.

What further examination can help reject the anthropic principle?