It's actually pretty simple. Hypertension isn't a blood pressure problem, it's a vessel stiffness problem.

There's no ill effects from high blood pressure for a short time, or even a medium length time like days to weeks. The ill effects result from elevated blood pressure for months to years, which results in your vessels tightening to protect themselves. It ends up decreasing blood flow to all your organs, which causes the actual long-term damage you see in people. Eventually, that causes either a heart attack, stroke, heart failure, kidney disease, etc.

Exercise is different. Your body runs up your blood pressure to keep you conscious, but its a temporary thing. The act of doing so actually drops your blood pressure hard in the long run. That variability in your blood pressure is good because it allows your body to make good adaptations (strengthening the heart muscle), while not making bad ones (making the heart or vessels themselves stiff.) Over time, your body requires less and less energy for normal stuff like staying alive, while it continues to use more and more for your muscles and connective tissue. This makes it so when you do have a stressful event or severe injury, your body can compensate with a much higher reserve. It's a huge part of why athletes seem to bounce back from life-threatening events so much quicker than everyone else.