If you are alarmed by the “typically much sicker” consider the context. These are observations in a hospital. The people being observed at worst also went to the hospital the first time or not at all, so that pool of “subjects” will be biased towards “sicker” a second time. People who do not go to the hospital a second time are not part of this observational “study.” They may have gone to the hospital the first time or not. They may have been completely asymptomatic the second time and not even know. Some may even have been asymptomatic both times. They cannot factor into the “study” thus biasing the results.
Most people who get it a second time are not going to be sicker than the first, particularly if they have been vaccinated, but even without. With the exception of cytokine storms (a very real problem with covid ... and some vaccinations), our immune systems do not normally work that way. On the other hand, our reactions to rhinoviruses, the common cold, varies depending on how we are abusing or maintaining our immune systems at the time. If the second time you catch even the common cold you are more stressed, isolated, sedentary, toxic, etc than the first you are going to have worse symptoms.