I'm working in an administrative area in a healthcare system so I can share my observation/experience which certainly can vary from place to place.
I have an MBA in healthcare admin which unfortunately was not AACSB because I didn't know what that was at the time. In terms of employment in my local area, having the graduate degree was all that mattered. Some employers offset years of experience requirements with the equivalent years from a graduate degree. I have not found it helps salary-wise compared to a candidate without a graduate degree, only in the sense HR might allow progression into the next level more quickly, again due to offsetting the years of experience requirements. Other than that, prestige or accreditation does not carry much weight here, at least so far, and this has been true through observing those in management positions as well. Years of experience is weighted MUCH more highly than degree program and there are several senior leaders and executives who do not have graduate degrees.
In terms of career progression, I have not found a distinction is made between these degrees. Having any of them will fulfill a preferred/required Master's degree for management or senior management positions. Some positions may specify "graduate degree in business or healthcare administration" (especially for something like finance, a MBA or MHA may be preferred) but more and more degree requirements are flexing toward a more holistic review of candidates. Once you're in a senior leadership position, networks and reputation seem to be prioritized and those leaders are often recruited for positions based on connections and experience.
The place where a Top 10 MHA may be especially helpful is in obtaining an administrative fellowship slot and interestingly that holds true even in a system that does not otherwise recruit many administrators with prestigious degrees. Top-ranked CAHME-accredited MHAs are the most popular option for admin fellowships, so that's something to consider. I do think administrative fellowships are a very valuable experience given the exposure and variety involved. Having worked in healthcare administration for several years, the internal path to a leadership position can be a long one without an organization dedicated to intentional leadership development and succession planning.
The admin fellowships can open a door for you. But that's not to say that finding an organization with strong internal career development or having a willingness to relocate to keep progressing is not even more valuable once you obtain those first few years of experience. While you learn a lot in a graduate degree, there is no substitute for working in the field and in that way, I do not think any of these degrees will prepare you better or worse for the real work environment in a hospital/health system.
That all being said, I'm considering going back for another degree(s) myself. A CAHME-accredited MHA could strengthen my resume and a CEPH-accredited MPH could add a different facet to my background and is of personal interest. I like the process of formal learning and having the credential brings a sense of accomplishment. Still, not every interviewer or organization is going to value that highly. Some just want to know you will do a good job and don't care about future aspirations to become a thought leader, moonlight in academia, or become a CEO.
There are managers and individual contributors in my organization with all three degrees, MBAs being the most common. As you might expect, MPH is more common on the operations side, MBA on the finance side, and MHA in general administration. But switching it up based on your interests or getting multiple degrees can make you a unique candidate and round out your skillset, especially if you aren't sure what you want to be doing.