Protecting the majority with the minority (old people, people with bad immune systems, etc) doesn't seem to be an ethical thing to do. (Think of this case. It's kind of like putting the pain of everybody onto a single innocent person. )
Essentially the idea was to do the reverse i.e. if the majority get infected -> gain immunity then if you can keep the (vulnerable) minority from being exposed while the infection->immunity process happens they won't need to get immunity because they will then be protected by the immunity of the majority (i.e. the herd).
In the same way that there are always going to be people who can't be vaccinated for a disease like measles (immunocompromised, too young, etc) but if a sufficient majority are vaccinated that prevents the disease ever getting an opportunity to get to them in the first place.
That means that there will be more people who are infected, so there will be more deaths.
If more people total get infected but less vulnerable people then you can still see less overall deaths.
Maybe do they mean to attempt to social-distance vulnerable people like old people but let other healthy people get an infection? Wouldn't that also overflow the hospital?
That was basically the initial UK plan - the idea being that if you could keep the spread amongst "healthy" people slow enough you'd "flatten the peak" and while the same eventual number would get the virus less would have it any one time, therefore (hopefully) avoiding an overload on the health infrastructure. This has sort of (although not entirely) gone out of the window now as it wasn't slow enough - and too many vulnerable people were getting infected.
The difference between this so-called "mitigation" approach and the China-style "suppression" approach is that in a suppression you're aiming to reduce the number of people that each infected person can subsequently infect to less than 1.
The change in tack to more intensive measures was driven mainly by modelling done by a team at Imperial College London, where they predicted that at the rate the virus was spreading the NHS would be overwhelmed and concluded:
We therefore conclude that epidemic suppression is the only viable strategy at the current time.
Hence why more drastic lockdown measures are now being implemented. Suppression brings its own challenges - namely that if it is successful the population at large remains non-immune and when you ease the suppression measures off too soon and you get any new cases introduced into the population it can spread very quickly again and you're back where you started.