If an ICE is, say, 15 minutes late, they cannot just drive faster to catch up. The schedule went on, and at that point there may be a much slower regional or intercity train on the same trajectory.
This is why ICE delays tend to cascade. It starts with a short delay, the ICE gets stuck behind a slower train, increasing the delay, etc.
The solution is better maintenance of tracks and trains, adding more rail capacity, adding redundancy, etc.
Of course, these are all much more expensive than an ICE speed experiment for PR.