Good points, but this is not something you can solve with a recipe. Investigate, talk to people and make sure you are solving actual problems and prioritizing the right tasks.

This is an extremely crucial step that you must do first: familiarize yourself with the system, its uses and the reasons it works like it does. Most things will be there for a reason, even if not written to the highest standard. Other parts might at first sight seem very problematic yet be only minor issues.

Be careful with number 4 and 5. Do not rush to fix or rewrite things just because they look like they can be improved. If it is not causing issues and it is not central to the system, better spend your resources somewhere else.

Get the team to adopt good practices, both in the actual code and in the process. Observe the team and how they work and address the worst issues first, but do not overwhelm them. They may not even be aware of their inefficiencies (e.g. they might consider complete rebuilds as something normal to do).