I am an IT Business Analyst, previously a Process Improvement Analyst in the business side before moving over to IT.
We mostly recruit internally from people who have a good range of our business knowledge depending on the area, supply chain, logistics, finance or procurement etc and have some experience of projects involving systems. Some of our analysts are ex-warehouse, export order processing, or production/material planning that have shown aptitude in systems and projects. If we don't have anyone internally interested it is difficult to recruit the needed supply chain/finance etc knowledge and business analysis skills externally.
Knowing the business and understanding processes (business and system) is more important than knowing coding etc, we have IT centres with who will write/sustain/provide ongoing support of the code from our requirements. Then we support the business and ensure they test the solution meets their needs. There is nothing more infuriating than business people writing their own databases/code, it becoming embedded in processes, then disappearing leaving no documentation and no one with skills to support and IT is looked at to sort out the gaps and mess left behind.
Like most business analysts we get pulled into BAU issues regularly, especially critical ones and that can be frustrating and stressful at times.
We don't get to look for pure improvements often which is what you seem to enjoy, we are assigned to projects. My current project is a 5 year global analysis of requirements and gaps for a new supply chain system we are deploying globally. It's the first time I have done this in something so big and where Agile is used and it is stressful, especially when finding out processes and scenarios in some regions I have not worked in before can be like pulling teeth.
If you are looking for something in IT Business Analysis getting involved in processes, systems, projects involving systems and having a look at the BCS courses would be a good place to start. Most business analysts ime can't code, that's a very different job.