I don't know if I can definitively answer your question - because IMHO it basically depends on how your organization wants to set it up.
Let me just say this: I have worked both in a multi-tiered organization as an institution admin where there were multiple admins or at least one primary admin at each facility, and I have also worked as an admin on a small team of 5 (most working part-time) that served more than 20 separate organizations of varying sizes. There are pros and cons to each type, but my preference would be to have a small team of dedicated admins, whose only or primary job is HealthStream.
If you have a dedicated team, you can afford to keep them well trained and they will have broader HealthStream skills. A handful of administrators can serve many facilities minimizing the number of people you need. If you are a small organization you may only need one admin, or maybe two. The larger the organization and the more varied the needs, you will need more people. The team approach means that team members may be able to specialize in an area (like maternity services or DI, or authoring, checklists etc.), but others on the team can cross train so that there is always coverage if there is someone out sick or on vacation.
What I have seen for the "each facility has their own administrator" model is that HealthStream frequently ends up being one of many responsibilities they have so they do HealthStream work as they can fit it into their other responsibilities. I often hear from some admins that they only use one small portion or function of HealthStream rather than utilizing the increasing spectrum of tools available to us. I often feel that way myself - that I have so many other duties, I don't get to play with and get skilled with some of the new tools. It's easy to let your skills slide or forget a process best practice when you don't use it often. It's kinda like buying a baby grand piano and only using a few of the keys to make music.
Regarding content development:
Again, I have seen both sides and the middle of the spectrum - from organizations that create most all of their content internally, and other organizations that use primarily HealthStream content and/or pre-developed content from one of the HealthStream partners. I've uploaded locally created video, facilitated voice overs and closed captioning. I've assigned pre-developed content. It all depends on what your organizational budget is - do they have the money to pay content developers if their own admins are unable to do that? Or, is there already content out there that meets the need? When budgets get tight, where will the HealthStream budget first get cut and how will that impact your training process?? All things you need to look at and think about.