Sometimes? The best analogy I read was to think of a car. Infectivity is the driver. Severity is in the car as a passenger. Other aspects are passengers. The ability for the virus to continue relies entirely on its infectiousness. Severity, as the front seat passenger, has a large effect on the driver, but the infectiousness, as the driver, is still in control. If the severity is high, the patient won't have much of an opportunity to spread, regardless of how infectious the virus may be.
Think Ebola or Marburg vs. HIV. Ebola and Marburg are extremely severe and show obvious symptoms very soon. As such, a patient infected with either will tend to show symptoms - and possibly die - fairly soon. Meanwhile, HIV can linger for many years without showing symptoms, giving the patient much opportunity to spread it. HIV is not as infectious as ebola, however HIV has infected orders of magnitude more people than ebola or Marburg.
Then, to further complicate the question, people acquire immunity. Getting vaccinated is similar to being infected, as far as your immune system is concern. And being vaccinated and/or infected will reduce the severity (usually) as your body is better equipped to fight it off. Also, some infections/vaccines offer a long immunity that effectively last a given person's lifetime.
So yeah, great question with a very complicated answer. Yes with a "but". Or No with a "well maybe".