Is that one-size-fits-all shaming solution really adding anything helpful to the topic?

Exercise is important, there is no denying that, however, the effect on blood pressure is usually not clinically enough. Even in the study you cited the mean drop in systolic/diastolic was -4.8/-3.19, which is small. The study in this post uses a criteria of -10/-5 to barely squeak out statistically significant results and demonstrate a negative correlation with dementia. Furthermore, according to the CDC about half of those with uncontrolled hypertension are at 140/90. Exercise is often inadequate to control hypertension and medications are necessary limit morbidity.