John Nauhaus' extensive comments in response to your related question apply here as well. However, there is a direct question that needs a direct answer and an emotional core to your posts that deserves more direct attention as well:
[Could] the graduate school administration change [my] graduation dates?
Is there a valid reason the graduate school would be unable to back-date my degree?
Your graduate school's administration cannot and will not change your graduate date if your request does not come with the support of your advisor and your department. The graduate school and the university may be "higher ups" in terms of a traditional administrative hierarchy, but in academic and curricular matters your department has absolute primacy. Any effort by the administration to "force" a graduation date on your department would be seen as an encroachment on their academic freedom, and that would get the attention of outside faculty and peer institutions in a tremendously negative way.
This may be unrelated to the harm that has occurred to you, but it is a valid reason for the graduate school to deny your specific request. If you still have any credibility within your university's administration, you could get more traction seeking other forms of redress.
Subtext: Does the department get away scot free?
You will not get what you decided to ask for. This does not mean that your efforts have had no impact. They have cost your department in at least two ways: First, to the extent that you still have the sympathy of anyone in the administration the dispute has cost your department credibility and administrative reputation. As you press the matter in increasingly intrusive ways, this cost will decline and eventually flip into sympathy. No program wants to be pitied, but it's better than being disliked. Second, if you've reached the point where a dean refuses to meet with you then you've earned the "problem child" achievement regardless of the merit of your complaint. Your graduate program admitted you, and will therefore be seen as bringing a "problem" (that would be you) to the university's doorstep. The administration—even the sympathetic administration—will call your department's judgement into question because they vouched for you.
Your department has paid and will continue to pay for what you went through.
None of this will be visible to you. That is what professionalism looks like from the outside: calm seas and a gentle wind, nothing happening here.
More subtext: Why should the department's error cost me?
This is the most important part of your question. As a student and teacher (can't and won't speak to this as an administrator), I have seen graduate students throw away their professional futures over pride and pocket change. It may not look like pocket change from the perspective of a graduate student's meager income, but you need to review your relative costs in terms of a tenured professor's considerable income.
Based on what you have written here, it looks like you are burning your professional future to ash over four years of lost pay. That "problem child" tag you've earned is sticky, and it will follow you. No faculty wants to hire a colleague that brings trouble, and no faculty wants to hire a colleague that escalates trouble.
Here are two ways to overcome this tag that I am aware of:
Be at the very top of your field. If your scholarly stature outweighs your immaturity, some good university will accept the latter's cost. This is emotionally easy—everyone would happily be the best at what they do—but it is intellectually difficult. Are you capable of being that good? What would achieving this cost you elsewhere? (It will cost you a lot. For example: Ever looked at the divorce rate for professors?)
Demonstrate a "newfound" maturity. If it looks like you learned from the experience, this will slowly offset the negative reputation you've acquired and allow faculty search committees to review your applications on scholarly merit. This is easier intellectually, you just need to be an employably strong scholar. This is very difficult emotionally because it requires an extraordinary humility. That's uncommon in academia; it can work against the strength of will and self-confidence needed to succeed as a professional scholar.
Deans, directors, and other professors may sometimes look like colonoscopy bags full of "pomp and disregard," but the attitudes that give you this impression are the same ones that got them through a dissertation, tenure probation, and other indignities of the profession. You seem to have that attitude in larval form, and it seems to be acting out. These "dismissive" deans and directors have learned to channel their attitudes into socially acceptable forms, and search committees will be looking to hire only those professional scholars who have learned to do the same.
John Nauhaus' comments elsewhere are going to be more valuable overall. The answer to your question about what the graduate school can do is little more than a footnote to his discussion of what you can do, and of what you need to do to recover personally.