No, that's the first option - it will eventually infect the whole world no matter how much you improve outcomes or slow it down. Unmitigated R0 of SARS-CoV-2 is extremely high, we can't keep it contained with spreaders all over the globe creating micro clusters every day. I didn't say it'll kill the whole world, just infect.
Many countries have already brought their case rates down to almost 0, and every single one that attempted to resume economic activity saw the case rates spike again, with many now re-introducing distancing measures again. This genie is not going back into the bottle.
The first option is actually potentially more viable on a reasonable timeline than vaccination, especially if we manage to find some kind of effective therapeutics which reduce the ICU and mortality rates.
There is however a potential issue, which is based on SARS data, immunity in recovered individuals only lasts 1-2 years. So unless there's a concerted effort to eradicate via immunisation, there's a measurable risk that SARS-CoV-2 will become a seasonal repeating wave like Influenza.