I agree the statement doesn't make much sense OP, obviously following the rules won't get rid of the virus. If everyone did follow the right rules (I don't know enough to say what those are) and if we also had effective test/trace/quarantine policies, it would help to bring case numbers down and that would allow us more wriggle room in what we do and what restrictions we can ditch. Obviously not allowing us to go back to normal since that would land us back at square one.
I am not of the 'let it spread' mindset as personally I believe that would see greater damage to the economy, education, all areas of health etc than the alternative. It would have dire effects on everyone, not just those who developed severe covid or died. If I think through the consequences of that spread realistically, it is not preferable to other approaches (not saying what we now is the best approach), not from any perspective.
Depending on the speed with which a vaccine becomes available I also don't think it would necessarily be over quicker. Listening to experts there seems to be a very widespread confidence that we will have a vaccine of some kind rolling out from next year. I think people do overestimate how much a vaccine will change things in the short term....it seems likely that many measures will persist while the efficacy and duration of protection are studied further, but in combination with other things it will make a big difference.
To me, both the statement about 'the more we follow the rules the quicker we will get rid of the virus', and also the idea that we can just let it spread and relieve some of the pressures that way, fail to see the wider picture or really think things through. I don't claim to be any kind of expert or know what is the right way forward, but I am fairly confident that those two approaches are both not the way, not as things stand.