Yes, it does. Obese parents fairly frequently have obese kids so they are directly affecting their own kids/family and setting them up with bad life time habits. Obesity is having a direct impact on the NHS services/budget for example. For example, Diabetes costs the NHS 10% of its budget, about 1.5% of the 10% is due to non-preventable Type 1 whereas 8.5% is for Type 2 Diabetes (as per
https://www.diabetes.co.uk/cost-of-diabetes.html). One of the major risk factors for Type 2 Diabetes is being overweight. Then chuck in Cancer as in the original article, specially adapted equipment needing to be provided, bed blocking as morbidly obese people need specialist social care/recovery beds causing other people's care being cancelled etc, so it is directly and indirectly affecting other people.
Also, the argument was about shaming causing mental health issues well I'm sure shaming somebody for smoking, taking drugs etc would have the same mental health issues but that seems to be acceptable to shame them. I certainly don't see much, if any, outrage when they get shamed.