Firstly, I’m sorry to hear that you were bullied in school and that it had such negative effects on you. I can certainly understand why you might not want to discuss that issue with potential colleagues at a university you are applying to. I cannot offer a perspective on what it is like to submit a diversity statement as a minority applicant, but I can tell you a bit about diversity statements, what they are for, and how a selection committee might view your position.
Contrary to the other answer given here, the real purpose of a Diversity Statement is not any improvement in the quality of teaching or research. That is the “brochure explanation” of its purported purpose, but the real purpose of these statements is primarily to act as a political litmus test to minimise hiring and promotion of academics who dissent from the neo-Marxist orthodoxy of modern academia (for discussion, see e.g., here, here and here if interested). As you correctly point out, diversity training and diversity statements have a poor record in bringing about any positive change in institutions with respect to interpersonal behaviour between people of different races, sexes, etc., but that is because that is not their purpose — their purpose is political/ideological, and they achieve their actual purpose quite successfully.
It is extremely tricky to try to predict the reception of a diversity statement, primarily because there remains quite a bit of heterodoxy —despite the goal of weeding this out— in relation to how academics view these. At the risk of oversimplifying, you might expect to get two kinds of main views, plus variations of these occurring in the middle. On the one extreme are academics and diversity administrators who find the statements desirable as a means of imposing a political litmus test on applicants, and they will favour applicants who express the correct neo-Marxist orthodoxy that they hold (the more radical the better). On the other extreme are academics who think that the entire exercise of diversity statement is illegitimate/useless and they will tend to give them little weight or ignore them completely. In the middle you will find various academics who try to square-the-circle, acting under the pretence that they can judge the diversity views of an application without any political or ideological favouritism.
As to your particular circumstance and your attitude to these statements, that is perhaps an unusual position, but not an unreasonable one. The advocates of diversity statements flatter themselves in thinking that these are a tool for helping minorities, so I suspect they would be inclined to respect your wishes not to discuss information that is sensitive to you due to your minority status. The other answer here gives good advice insofar as it notes that you can concentrate on output-based issues (e.g., your teaching, research, etc.) and ignore your own personal characteristics if you want to. On the other hand, ideally you should be able to speak to your actual views on diversity exercises, and you ought be to able to use the statement to mount the anti-diversity-statement argument you have given here. The very nature of the statement as a political litmus test means that the result of this is unpredictable, and heavily dependent on the ideological and political views of those who read it. Since your view is that the statement is window-dressing and that what is needed is “structural” change in the university, that would probably put you on the desired side of the ideological divide that the litmus test is designed to elicit (arguably more radically so), so you would be unlikely to face negative repercussions from taking this view; depending on who reads your statement, it might even count as a positive.
One day, perhaps we will all be mature enough to judge an applicant entirely on the quality of their work. Until that day comes, you may have to resign yourself to being one of the many applicants whose employment prospects hinge on unobserved ideological and political discrimination. Hopefully this answer is useful to at least help you understand the position you are in.
Update: You might also be interested in this: FIRE statement on the use of DEI criteria in faculty hiring and evaluation.