I critique government a lot, and think we have gone overboard in our coronavirus lock-downs. But let me be somewhat more generous to government decision-makers for a moment.
There is a wide range of policy options between "do nothing" and "lock down the economy as much as possible to keep people at home." Government calling for purely voluntary social distancing is itself a policy option, even if it means government takes no active enforcement measures. And of course some of the non-do-nothing/non-lockdown policies proposed are to ramp up testing more rapidly and relax regulations on medical equipment producers.
So that means government has options, but part of what government has to consider is the signalling effect of their policies, not merely the direct effect. That is, government needs to consider whether its policy options persuade people to take the danger seriously or whether they inadvertently signal to people that it's not that serious.
Logic isn't the primary issue here. We shouldn't assume people make the effort to carefully parse the full implications of what the government is saying or doing. They'll respond emotionally, or with superficial consideration, or with a heaping dose of confirmation bias, etc.
That being so, policies that don't appear to signal a dire threat are problematic. They might work well if everyone took them seriously, but people might not because they don't signal seriousness. On the other hand, lockdowns at great economic cost signal seriousness.
This applies even to policies that don't directly involve the general public, such as reducing regulations so more tests, ventilators, masks, etc., can be produced quickly. That could be done without too much public notice, and it would have good effect, but not maximum effect - it would increase the needed medical supplies but not change public behavior. However, if government relaxed those regulations, and made a big noise about it -- "this is a time of such great crisis that we have to act boldly and accept the risks; therefore we are temporarily relaxing our normal regulatory procedures to try to save tens, maybe hundreds of thousands of lives" -- it would both make more equipment available and send a useful signal to the public.