Should I / do I have the right to tell her she needs to acknowledge and handle her responsibilities as a parent or just leave it?...
Just leave it alone. You handled it the right way. She is going to try to dump her child whenever she's too tired, he's not minding, she wants to go out and play with her friends or OM, etc. Don't rescue her. We still have to be parents when we are blue, struggling with life, etc. This is the reality of having a divorce! This is what she has to "get" in her head.....that divorce was not her best option.
I don't know the extent of her depression issues, if she was suicidal or may endanger her son. Is that a legit concern....or is it your protective instinct talking? Otherwise, I completely agree with the statements from Darkness. This is a critical stage for her, and I don't think you should constantly tell her you are there for her. That is what you tell a spouse who is in a MR with you.......and she tossed that away. Not as punitive means, but you need to get out of the way and let "life" show your W the consequences of her decisions.
I would say this new lady you have met, has contributed to you feeling more detached from your W. Be very careful and don't complicate your life any more than it already is. You may be emotionally vulnerable to any new person who makes you feel good about yourself. I should know, b/c that is how some affairs begin.
So, step carefully my friend, and don't be surprised by the reaction when your W hears you are dating. Actually, the timing couldn't be better (not that I am condoning it), just saying...
It is not about what you feel should work in your M. It is about doing the work that gets the right results. Do what works!