How to drive our career towards the role of program manager from being a SDE and some experience in project management?

Hi, good question. If you have experience in Project Management, you know the basics (achieving the objective by managing Scope, time, quality, cost, resources, and others). You can understand it better in the analogy of a guitar player and the conductor of the Orchestra. As a project manager, you are a guitar player, you can play the guitar very well, that is one part of the orchestra, as a Program manager you have to manage every piece of music played by different instrument players.

Now the question is “should the conductor know to play all the instruments” answer is “no”. But he must have an excellent idea of the final piece of the music he wants to create. As a program manager also, you must know what is the final objective you want to achieve by managing all the different projects running in parallel and making sure they run in synchronization.
I wish you the very best.