I can say that I didn’t used to bother to cool down and didn’t have any problems.
This last time I started exercising again (I’m not very good about it), I’ve been walking on a treadmill. Not particularly strenuous, not for tremendously long stretches (30-40 mins), and I’d do a minute or so of slower walking at the end to finish.
Kept being dizzy and lightheaded when I got off the treadmill. Asked my doc, he said to do a real 5 minute cool down to let my blood pressure drop more slowly.
I found one theoretical concern anyway: the active cool down period is supposed to help prevent venous pooling and some think that preventing such may reduce the risk of having “an occlusive event.” Theoretically.
The present consensus is that one should not stretch before warming up. When I began running, we runners would all stretch for 10-15 minutes before the run. Then the scientific establishment stated that not only that did not prevent injury, but can actually injure cold muscles, ligaments, tendons. The present advice is to stretch after a short warm up. So, yes, warm up first, but stretch only after the warm up.