I say it's wrong because months ago I heard arguments that models predicted that lowering the curve could also reduce total mortality. And I linked to that video to give evidence that this was part of the general discussion months ago.
The fact that you disagree with that model, or argue that I need to give a stronger evidence for the current information we have, has no bearing on my point.
Eg, when you write "I don't believe this is the situation right now" - that statement is based on knowledge you have now, not knowledge that people had on "day 1".
Nowadays it seems both EU & US hasn't pursued heavy suppression (probably too costly, and legally and culturally difficult), but rather just hoping to keep it somewhat under control until a vaccine comes, hopefully.
I believe it's the burden of the proof is on you to present a new, more rigorous model if you want to argue that slowing down transmission significantly lowers overall mortality - as the null hypothesis is that they are not associated.
On the last point, I am aware that under the SIR model, there's the possibility of overshoot, i.e. exceeding the minimum herd immunity threshold. My understanding is that this isn't a significant effect unless you have uncontrolled, explosive expansion, so that most people get infected simultaneously before others develop immunity and can therefore act as a barrier for herd immunity. I don't believe this is the situation right now given the numbers we are seeing.
That said, I believe that the more realistic future is that a vaccine or an effective treatment will be developed in the next months, and that will be the thing that will have made temporarily lowering transmissibility worth it. What I don't believe, because I haven't found any evidence for it that holds basic scrutiny, is that if no vaccine or effective treatment can be developed in a reasonable timeframe, then keeping the measures to slow down transmission as long as possible will have any significant effect on lower overall mortality.
EDIT: To make it clear so people do not misunderstand me, what I believe has no evidence is that without a timely vaccine or effective treatment, slowing down transmission beyond what is necessary to keep the health system from overloading is associated with lower overall mortality.