Don't think of these as cons, but more as facts to consider before taking a job at the IT department. If you're a typical business analyst that thrives as a change agent on creating or modifying systems to achieve growth in productivity, this may not be the right place for you. All of their systems are commercial of the shelf (COTS) products. There is virtually no wiggle room for product improvement because you are at the mercy of the software vendors. You can put in a ticket to request something, and maybe you might get it, but most of the time they don't. And if they do agree to make the change, you have very little influence in how it's done. A handful of important systems are old and archaic. Users will ask you for information that's nearly impossible to retrieve without spending at least a whole day working on it, and it's literally very manual copy and paste efforts. Becoming experts on some of these systems won't take you far in your career. Your day at work will also be entirely tied to help desk tickets. You'll primarily be a software administrator. You won't be putting together specs to do create better things. They simply don't operate this way.
Don't misinterpret the job as not important. It definitely is; the hospital needs all the help they can from IT to keep patients happy, but the work itself doesn't lead to much growth. The job is literally only support...there's very little in the job for improving systems and processes. The only area where you might see some decent opportunity for improvement is the integration between all these COTS systems, but that area is already stacked and very hard to get into. In the greater context of what business analysts do at Fortune 500 companies, the work here is definitely of lower tier. I don't say it as an insult...it's merely a fact.
Although to be fair, most of this has to do with the small budget that IT is given to work with, and I suspect the leaders of this hospital are doing the best they can. So just take all of this into consideration before making a final decision.