Very interesting. I was going to make the same suggestion. Specifically bouldering since I think it's generally easier for folks to plug into their daily routines with the proliferation of gyms and not needing a significant amount of equipment. But three times per week bouldering has radically changed my body and hands for the better. I no longer have elbow or forearm/wrist pain that I used to experience quite frequently after typing for a long period.
I also want to mention that I am in my 40s and had never climbed before I started about 5 months ago. I just signed up to take an intro to bouldering course at a local gym and went from there.
Additionally climbing is a very technical sport that involves a lot of problem solving which I find very appealing.
Being overweight or out of shape shouldn't stop you from giving it a try, you might stay climbing lower grade problems longer than others but you will still find challenging physical and mental puzzles that you can complete.
