It shouldn't be a goal. The virus cannot be stopped with our current medical technology, but you can certainly punish people into poverty and homelessness with draconian lockdowns.
You say it is not possible, but it absolutely is as it has happened. There are countries in this world which have had community spread and then stamped the virus out entirely, like Taiwan and New Zealand have. There are countries which have been in as bad a situation as we are (like Australia and South Korea) and then gotten their numbers down to the point where TW/NZ were at their peak. So there is a pathway from rampant community spread to eradication.
> These countries you mention are vastly different in terms of population size and geography.
Yes, South Korea and Australia span the gamut. So do Taiwan and New Zealand. The USA is somewhere in-between.
> Also, pretending that there are no costs associated with strict lockdowns is terribly flawed.
Those costs are largely suffered by this lockdown-reopen-lockdown cycle, without the benefit of getting rid of the disease. Indeed our trajectory is worse economically, because there's no light at the end of the tunnel.
Life in Taiwan and New Zealand goes on pretty much uninterrupted right now. We could have had that. We still could, if severe actions were taken.