The significance of peak period in surface wave models
The peak period is only one of multiple parameters that is used to describe the spectral shape of ocean waves. Most commonly used parameters are significant wave height, mean period, and dominant (peak) period. Significant wave height is proportional to the square-root of total wave energy and it tells us how much bulk wave energy is there at any given location. Mean period tells us how long are waves on average, and peak period indicates how long are the waves with maximum energy. These are just statistical parameters describing the shape of the spectrum. In reality, accurate prediction of the whole spectrum is important for wave prediction.

What happens if we vary the peak wave period while keeping the total wave energy, and thus significant wave height, constant? These are some of the effects that can expect with longer peak period:


Swell propagates faster and reaches the shore sooner, as you mentioned in your question;
Because swell propagates faster, wave dissipation from swell outrunning wind will increase;
Longer waves are more dissipated due to bottom friction, i.e. longer waves will feel the bottom sooner as they propagate toward shallow water;
Longer waves induce Stokes drift that decays more slowly with depth compared to that of shorter waves, inducing stronger Lagrangian transport at depth and enhanced Langmuir circulations;
Longer waves with same wave energy are less steep, effectively decreasing the dissipation of short waves that "ride" on the forward faces of swell waves;
Different interaction with structures, as mentioned by Wolfgang Bangerth.


Because of the above physical processes, waves at peak period affect the evolution of the spectral shape both locally and along their propagation line.