FWIW I agree with your interpretation.

To take it further: McGonagall accompanies Harry to Diagon Alley, while (on Dumbledore's orders) learning as much as possible about him. She attempts to report to Dumbledore, but is speechless; Dumbledore may or may not be reading her mind, I don't know. Before this, he was happy to delegate the responsibility, but after this meeting, he naturally decides he'd better investigate personally. Platform 9.75 is the next convenient opportunity.

Yes, that's a lot of detail, but I think the story bears it out.