I know the President suggested that but there doesn't seem to be much scientific basis for the claim. I hope it's right but I'm not going to count on it until there is some backing evidence that the statement wasn't just wishful thinking.
I know this will probably generate some "lamestream media" comments but this piece illustrates the projected results of different timings pretty clearly. Before you completely discount it based on the source, consider that the actual data and model comes from epidemiologists.
The biggest issue I see with the simplified model is that the ICU capacity is not included which, if it were, would dramatically increase the number of deceased people. The moment that ICU capacity has been reached (and don't forget there is a regular demand on ICU capacity, e.g. accidents, operations, heart-failure, etc.) the Death rate will skyrocket. Also, the capacity would be under pressure from ICU workers falling ill.
Also, no, not everybody who gets contaminated will die, but probably between 1% and 3% will, and it's not only older folks, although they do run a higher risk due to reduced resistance and the likelihood of having additional deceases which makes them more vulnerable. Increasing the testing capacity will allow to reduce the Deat Rate somewhat, but we're talking about millions of deaths.
So, the scenario that sane people are following is to apply Social Distancing to avoid overburdening the ICUs and staff which, assuming people adhere to the guidelines, can then be slowly relaxed to
manage the pressure on ICU capacity. This will gradually build some group immunity (early testing suggests that immunization is happening), and it will bridge the time to develop a vaccine (in a year's time) in large enough quantities that would allow relaxing Social Distancing more rapidly.
There is one simple recipe to disaster, and that is to try and speed up the economic recovery (for electorial or whatever other selfish/misguided reasons) by lifting the social distancing to fast. Brazil will be a sadly interesting case to follow.