NTA you are doing good by your daughter. I just wanted to point out that no one loses weight from 30 minutes of running a day. Your wife has nothing to worry about in that regard and when she is overweight herself and presumably through her fair share of diets, she knows this. So this is really some deeper emotional level you two have to tackle.

She might feel indirectly criticized for her weight. She might be jealous that you have bonding time with your daughter while she seems not to have a similar activity with her. She might not actually prefer to stay at the empty house but struggle with your walking speed and doesn't want to admit it (you know better how fit she is). She might be reminded of failed attempts to start exercising herself.

There are a bunch of possible issues here that need to be unraveled to get you guy back to the "we against the problem" mindset.