Why is a damaged bike tyre a problem?
Two things could happen:

You pump up the tube, and after a while, the side of the tire yields. Usually when you´re sitting on the bike, because that increases the pressure a bit more. Looks like it has the plague or something, develops a well visible bulb. The bulb grows, and after a few moments or minutes poof (or bang if it´s a racing tire at 8 bars) you´re out of air. Can happen at a dangerous moment. Usually you hear a funny sound of the bulb touching your frame before and have time to slow down.

You brake, and the side of the tire disintegrates under the sudden stress. Perhaps more likely with lower pressure (otherwise (1) occurs before). Complete loss of control if it´s the front wheel.


Both of these only happen when the carcass of the tire is damaged, due to riding on a flat (likely invisible), a maladjusted brake pad, or something else that has been making quite some noise while rubbing off the rubber down to the carcass.
Visible cracks in the rubber can grow for many years before the reinforcing fabric below is destroyed. If you bend the tire, and pieces fall off, yes, then it´s had it. The crack growth is sped up a lot by riding (or even letting the bike stand in a dry shed) with too low pressure. Rubber that is bent + ozone from the air = environmental stress cracking.