I believe Denmark or some other country in that area of the world is trying exactly this. They are not social distancing and are going for herd immunity. I don't think they had a big problem at any rate, but the answer is probably, yes, theoretically if herd immunity is actually possible with covid, and we don't actually know yet much about whether or not having had this disease confers immunity, there's mixed theories on this and apparently many people who have already been infected more than once, though not most by any means, so keep in mind that immunity of any kind is still a theory, not a fact. But the cost will be millions dead. Do you want that? Also keep in mind, this virus is much like the common cold and we've never developed immunity to that.
We've never acquired immunity to the flu. Immunity isn't a guarantee. As to bats, the reason they live with so many viruses is a very very strong immune system against them, it has nothing to do with social distancing. On the other hand, there's a mold I believe that is rendering bats endangered, so that immunity to viruses isn't an immunity to everything. One has to understand, when any species becomes too numerous for the planet to maintain, this kind of thing happens. We have been able as humans to fight it because of our development of a cerebrum, but that hasn't stopped diseases and our own violence from culling the herd. Right now we're killing off the favorable climate that allows us to exist in such large numbers, so again, if we don't react it's very likely that will lead to an even larger culling. Everything dies, and death makes life possible. None of us wants to volunteer to die for this so nature comes up with something and then we try to outwit it. We could fix it, perhaps, with birth control but that concept came pretty late in the human population explosion. So with bats, despite not getting this virus they do have predators and do not live forever.
Apparently it was discussed in the White House and in Great Britain, but not for very long once the speed of the spread of the disease became apparent. A lot of deaths all crowded into a relatively short time, from a source that wasn't there before and so are new amounts added to the usual numbers of deaths in a year, would stress a lot of systems or even cause them to fail. (I think that at the present projected 1% estimated rate it would be around 3 million dead in the US and 600,000 dead in Britain, but even if it was a tenth of that, at .1%, it would be 300,000 dead in the US and 60,000 in Britain.) The medical and public-welfare systems would also have to take care of many more who were sick and needing care. My guess is that those making the decision were concerned not just for humanitarian reasons, but also because the hospitals would be overwhelmed (like they were in northern Italy) or even that the carrying capacity of society (food supplies, etc.) might become overwhelmed if the projected deaths were not spread out in some way.