"What would be the most important thing that a new software manager or lead should do as he makes the transition?"
As if there possibly can be just one such most important thing that suits any newly promoted software manager or any circumstances. Bad question. However, actually, having been in that situation and having got burned I can confidently say, that there is such one, most important, the ultimate answer to the ultimate question, the thing any newly baked manager should do or rather not even attempt doing.
Just reflect on your past experience as a developer, what would be your golden rule, the ultimate and most important thing you should do (or rather not even attempt doing) when faced with the task of maintaining and improving a complex, highly coupled and awfully important system you do not really understand? Correct. If you do not understand how and why it works do not change it. Full stop.
Being a good programmer would you change a piece of any production code without fully understanding what it does, why it does that, what are you trying to achieve, how are you going to achieve it and how are you going to check there are no side effects? Doubtfully so, especially if you are really good programmer, not until you gain enough understanding and confidence that the change is safe and there is a contingency (plan B as they say).
Manager deals with a multifaceted system of people, resources, tasks, explicit working procedures and tacit practices, politics, statutory issues, company policies and the list can go on and on. The last thing you want is to break it, disrupt or bring to a halt. At first just let it work, do smallest possible changes, have a plan B, and see what happens, learn. All that good stuff any experienced developer would do when assigned to maintain and improve a production system they do not fully understand.