The only plan that would not involve herd immunity at all would be a zero covid one, something that most people agree just wasn't ever going to work in the UK.
Herd immunity involves a threshold. Unless you hit that threshold, which might be 75-80% of the population, you dont get herd immunity. Instead as more people get infected over time, you get a gradual reduction in the pool of susceptible individuals. The number of susceptible individuals in the population is one of the key variables that epidemiological models use, it makes a difference to the size of the wave projection curves such models generate. But this isnt the same as herd immunity, which doesnt kick in unless enough of the population are immune that others are protected by virtue of the virus struggling to find populations to transmit in on a scale that enables exponential growth. A threshold for herd immunity also relies on other factors such as how good the immunity is, how long it lasts, whether it stops infection and transmission.
Although the term herd immunity was sloppily used in a broader way, it shouldnt really be unless the policies etc actually involve trying to hit that final threshold. The much slower increase in the proportion of non-susceptible people that countries saw as a result of preventing many but not all infections shouldnt really be labelled as herd immunity. Indeed, when other factors such as strength and length of immunity are factored in, even after many millions of infections and huge vaccination programmes, the real version of herd immunity arguably still doesnt fully apply to this virus. Those who are still vulnerable to death from this virus are not being fully protected by the amount of immunity in the population, so deaths are ongoing. Obviously the scale of the problem has still changed a lot, which is why we dont have restrictions any more, but the pure version of herd immunity was never reached.
I don't think it was ever a binary yes/no to herd immunity being part of the plan. The "flattening the curve" approach makes sense when you have no idea when vaccination might be possible - unless you are going to impose quite drastic measures for an undetermined period of time.
Since the original plan was very much the standard UK pandemic plan, we can look at it from the angle of other pandemics (usually flu pandemics) too. In these contexts the population immunity picture has always been part of the rationale for letting the pandemic virus spread. An immune naive-population is what makes a virus have pandemic potential in the first place. Epidemiologists worst nightmares always involve the susceptible population pool being huge, and one way or another that pool will indeed decrease over time and we dont get to exit the acute phase of the pandemic till that happens. With flu the assumptions about vaccines becoming available within a reasonable timeframe are stronger. But traditionally there was also the assumption that the numbers game, in terms of hospitalisations and deaths and public acceptance of the situation, would mean the government could afford to let a big chunk of the population catch the disease over a series of waves. In the case of cthe covid pandemic, neither the virus, the public attitudes or the health system capacity enabled this approach to be sustainable. If they had stuck to the original plan for longer, they would still have been forced to switch to the suppression plan at some stage.
Hopefully this series of posts demonstrates that I am not simply disagreeing with everything you said, and the picture is messy with various degrees of success and failure, and other factors and ways to measure success.
As for zero covid, unless the whole world had managed an elimination strategy, there was no reasonable prospect of it being a permanent solution, and the few countries that went for a zero covid approach did so to play for time. The successes of their approach show up in both their health and economic data, and how long and drastic their lockdown periods actually were compared to ours. But they still needed an eventual exit strategy, one that was enabled by vaccines.
In many respects the difference between the policy we ended up having to go for had some key similarities to the situation the zero covid countries faced, just with more death in the meantime. Every country ended up in a situation where the fullest version of 'let it rip' was impossible, even if you imposed no formal measures the population changed their behaviour to some extent. Everyone was left needing an exit strategy that involved some combination of vaccines and some proportion of the population catch the virus at some point, with the biggest differences involving various sorts of timing. At certain stages of the first 18 months or so, what mattered most was how timely countries were in trying to bring R below 1, not whether they were then aiming to get case rates all the way down to zero or not.