> The same number of people will get sick but over a longer period of time.
Fewer people get sick for two reasons.
Flattening the curve reduces the amount of epidemic overshoot. People assume you reach herd immunity levels and the epidemic then dies out. Reality you overshoot herd immunity.
At some point you can reestablish contact tracing and isolation. That by itself reduces the required level of immunity and containment needed to keep the infection in check. Except for a few countries we don't have that now.
My bet at this point is within a few months. Rapid RNA tests will start being produced in large numbers. Synthetic anti-bodies for covid19 will become available. Contact tracing and isolation containment will be reestablished. Rapid RNA tests make contact tracing possible at scale. Antibody therapy increases the effectiveness of contact tracing. Basically you test everyone with symptoms. You then dose all their contacts with anti-SARS-COV2 antibodies.
Not the end of the world.