Following the Project management community since 2012, it is the first time I have read a question that describes many teams and many projects in such a scale.
In project management, what is the most important to my own opinion is to respect the deadlines and the budget. I take into account that every team you manage has a deadline and a budget.
Since I did my BAA courses in project management in the late 1980s, project management tools have evolved a lot. However, they seem related mainly to the IT industry and the construction world. Managers who have a degree in business administration learn other skills to manage a project team than how to use project management software. Even if many project managers don't like it and see that as over controlling, once a week you can call the project managers to see if they are going over budget and if their deadlines will be respected. If not, you can have a meeting with them to put them on track.
It sounds easy, but in IT and in the construction world, communication on those issues is a sensible point. The reputation of the project manager and of the contractor is at stake. So, having studied in business administration helps to manage susceptibilities; to increase the motivation of the manager and of their team members and it develops the quality of life at work.
You can use a word processor to do a weekly report that you submit to your own boss. When a manager demonstrates that he or she can provide those kinds of information without creating tensions or resentment, the teams he or she supervises will grow in innovation and determination.
Am I old fashioned, to believe that a telephone, emails and a word processor are sufficient to enable a manager to know really what is going on in his or her work teams? I don't think so. Since email appeared, many managers hide behind them, don't follow the budget spent and don't monitor the deadlines. So, the work environment is transformed into a competitive environment where some teams win and some teams lose; the usual blaming game starts when nobody is able to explain the failures that happened.
Too many times in my career, I've seen employees blamed for things they didn't do, just to protect the job and reputation of another. In a few years, that phase will be over and giving a phone call to all project managers will be perceived as normal work that builds a fun work environment to work in.
Those are general principles of management, but picking up the phone, sending an email, and writing a global report is the real test of competence in management.