Why does high blood pressure not damage the body during exercise?
Actually it can. See https://clinicalhypertension.biomedcentral.com/articles/10.1186/s40885-016-0052-y for instance:

Deleterious effects of HRE [hypertensive response to exercise] on structure and function of left ventricle (LV) has been reported consistently. Theoretically, individuals with HRE would be exposed to abnormally high pressure loads to left ventricle (LV), which may result in global subendocardial ischemia due to mismatch between demand and supply from excessive rate-pressure stress. Indeed, a previous study demonstrated a greater likelihood of new or worsening abnormalities of wall motions from echocardiography in individuals with HRE, even in the absence of angiographically significant coronary artery stenosis.

The issue is that you're confusing somewhat high blood pressure with abnormally high blood pressure during/after exercise, i.e.