Although sad, at least the European Union is ready to replace these with the CO2M constellation (originally Sentinel 7) to be launched in 2026. It will have a higher resolution, higher quality, and broader coverage.
I agree 100% with this comment -- you can clearly see the energy separating into its component frequencies as the solution evolves. The reason is that different frequencies of the solution propagate at different speeds. Smoothing the discontinuity would indeed reduce the high frequency components, such that the effects wouldn't appear so extreme. The shallow water equations only allow for one shock wave mode as far as I recall, it wouldn't explain why we seem to get increasingly more shocks as the solution evolves.