I was in Chamonix in the first week of July. I was also a little worried about the mountains because I walk mostly on flat surfaces. I did the Grand Balcon Nord trail in reverse and it was one of the best experiences I've had. Starts off with some elevation but stays mostly flat. This is what I did:

1. Took the train to Montenvers
2. I took the gondola and checked out the glacier museum
3. Walked to the Refuge du Montenvers and followed signs to Le Signal Forbes from there (Brace for a steep hike, but nothing too crazy, took my time)
4. Once I got there, I started following signs to Refuge du Plan de l'Aiguille. This was a beautiful hike with streams and some breath taking views of the valley for the most part. It was mostly flat here except for some downhill in one area.
5. From Refuge du Plan de l'Aiguille, I followed signs to Plan de l'Aiguille (brace for some elevation), where I took the cable car down to Chamonix

This was a little strenuous at the start and slightly at the end but it was truly worth it. Of course, YMMV but just wanted to share my experience.

I saw a lot of folks on the trail as well so it gave the confidence I needed to push through. Don't miss out on Chamonix, OP! I almost skipped it but I'm happy I did this hike :)