Why is a damaged bike tyre a problem?

A sudden failure of the front tire is very likely to cause you to crash. The tire can come off the rim or otherwise make you lose traction.

If you are lucky you are going straight at a slow pace. If you are unlucky you are going through a turn at 50km/h and there is a 100m drop at the side of the road.

A sudden failure of the rear tire is still bad but more stable and manageable.

Two things could happen:

1. 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.

2. 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.

Can someone explain the full consequences of a damaged tyre and whether it should be replaced or not?

Depends on the level of damage.

Things like scuff on the sidewall that don't penetrate to the carcass (reinforcing fibers), cuts in or chunk taken out of the tread are generically not a problem.

If the damage is something that could cause the tire to fail, that's a safety issue and the tire should be replaced. That includes any damage to the carcass, a hole that would allow the tube to herniate out of the tire or the rubber tread missing so that the carcass is visible and would contact the road, or damage to the bead.

A really damaged tire may puncture more readily, suddenly blowout, negatively affect braking or traction while cornering or come off the rim. Most of these things may cause a crash. If a tire failure doesn't cause a crash the wheel rim may be damaged.

Beside the obvious potential crash damage there is potential damage from the tire failure. If the tire rolls off the rim you could ruin the rim on the pavement before you can safely stop. There is the chance the tire could catch the brake caliper damaging that also.

As a very minor side effect, as a tyre wears and is damaged, the more rolling resistance it acquires.

Check out this https://www.bicyclerollingresistance.com/specials/grand-prix-5000-endurance-test where the sample tyres have added a watt of losses between new and the first 1000km checkin. After that losses still grow but more slowly.

I once damaged my tire by pushing the bike when the tire was flat. It was a quite new tire actually. One day, when I came back to where I had parked my bike, it was flat. Closer inspection revealed that the tire had exploded while the bike had been parked! This was in winter and under a roof, so absolutely not temperature stress.

The point is a damaged tire can explode at any time. You cannot trust them anymore.

Now, it's not a problem if it explodes while being parked. However, if it explodes while you are riding downhill through a corner at 60km/h, it'll get you at least a free ride into the hospital. Unfortunately, tires tend to explode when they are stressed in one way or another, that those situations are those where an exploding tire is most dangerous.

Your life depends on your tires. So ride tires that you can trust.

I get through two pairs of tires a year. My rule is to replace a tire if...

1. There is a hole I can see through.
2. I can see any part of the white nylon/fabric layer from the outside
3. It gets more than one unexplained flat
4. It has more than 5000 miles (for a road tire)
5. It has any bulges or a broken bead

If you don't replace a damaged tire you will eventually get a flat which may cause you to crash. Flats caused by damaged tires tend to be blow outs and cause serious crashes.

Note that the rear tire wears faster than the front. Some people will move their front tire to the rear and put the new one on the front. You don't always have to replace both tires at the same time.