A problem with armchair epidemiologists is that they can make very confident-sounding statements without acknowledging (or perhaps even recognizing) that those statements depend on assumptions. https://twitter.com/federicolois/status/1298631150316007428
The claim "the longer the path to herd immunity, the worse the total fatalities" relies on some very strong assumptions.
Before I get into that, let's highlight an observed fact - every single country has worked hard to prevent introductions and spread of infection in aged-care facilities, while also keeping community spread low.

No country has succeeded.
Will it be easier to prevent introduction and spread in aged-care facilities if community spread is high?
Let's go back to the quote that's triggered this thread:

"the longer the path to herd immunity, the worse the total fatalities".

This claim is dubious.

Fatalities don't depend on how long the path is, but how many transmissions reach the vulnerable group during that path.
There are some assumptions that I think are made here which I believe are dubious.

1) in normal society the vulnerable do not preferentially mix amongst themselves.
2) we can isolate the elderly while actively promoting a raging epidemic in the remainder of the population.
3) We can get to "herd immunity" quickly, and then turn off transmission
4) There is no difference in outcome for someone infected in 12 months versus now.
Why does this matter?

If aged-care facilities can support sustained transmission, then the concept of herd immunity without large numbers of vulnerable people infected does not exist.
I do not think we can keep infection out of aged-care facilities if community spread is rampant.

Here's a Fields medalist explaining the mathematical challenge in this: https://twitter.com/wtgowers/status/1259424414220931072?s=20

But, no need for theory - just look at the failures in practice so far.
Also if we slow the epidemic (take a longer path), we gain time to improve our treatments (so when someone is finally infected, the treatment is better).

Also, we get better at preventing transmissions (e.g., more masks), which reduces the spread and the herd immunity threshold.
And if we keep below the herd immunity threshold till a vaccine gets us there, then we don't need all those infections.
We need to also recognize that when we achieve herd immunity threshold, if we do so with a large number of infections present, then many more people will still be infected after we cross that threshold. (this is referred to as "overshoot").
I would advise we want that number as small as possible, rather than trying to get to the herd immunity threshold as quickly as possible (which will maximize the overshoot).
Before anyone argues in support of this "promote infection in the healthy while isolating the vulnerable" idea, he or she has to explain how we isolate the vulnerable.
To anyone who refers to Sweden on this (and @federicolois started his thread with a reference to Sweden), you do realize that they have prevented visits to aged-care facilities for 6 months so far, right?
[and by the way, if you're going to criticize me for some bias, you should check my preprints earlier in the pandemic, and you'll see that my initial research was about how to make this sort of strategy work.]
So I agree we need to have more serious work trying to understand the harms of lockdown and how to mitigate them or replace lockdown with better strategies.
But I do not believe that actively promoting infection of apparently healthy people while attempting to isolate the vulnerable is going to do anything but increase deaths in the vulnerable.
And again, I've seen many people say "we need to maximize infection in healthy people while isolating the vulnerable". They've had good suggestions on how to maximize infection in the health, but nothing about how to isolate the vulnerable other than "isolate the vulnerable".
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