I think the problem is that household mixing is exactly where the most transmission takes place and is the most difficult to control. Social distancing is pretty meaningless when you get a family or close friends indoors. Our brains just aren't wired that way.
However, what's not clear is how much transmission is from households mixing and how much is within households. I suspect that the majority is the latter with not all that much of the former.
That's not to say that it's not something that helps reduce numbers, rather when comparing it against (for instance) going to the pub it could well be that pub and household mixing could be similar but the official figures include within households and so pubs look a lot "safer".
Unless there's data available which breaks out down more.
The 25% reduction will make a difference, but when ICUs are running at double their usual occupancy in places (L&SE particularly) it's still a major issue. Between lower case rates and increasing herd immunity effects (although vaccinating the elderly probably won't have much impact on that of itself I would have thought) there should be some cause for optimism, but I don't think we'll start rapidly dropping restrictions in mid February.
Vaccines will only work if cases are low or nearly everyone has had it. Almost anything else would have a small impact without other measures (such as the current lockdown).
You also have to remember that from vaccine to not going to hospital can be about a month (at least 2 weeks for the vaccine to prove protection and then 2 weeks before they would have gone to hospital). Therefore although some (but probablya carry small amount given the limited number vaccinated 2 weeks ago) of the current falls in cases can be associated with the early stages of vaccination there'll be almost no fall in hospital cases yet (due to very few having been vaccinated a month ago as there'd only been 10 days of vaccinations at that point).
Therefore if we got the mid February target that's not going to show up until mid March on the hospitalisation stats (although between now and then numbers will start to fall due to the lockdown).
Without the lockdown is likely that a 25% reduction due to the vaccine would have been offset by rising case numbers.
Whilst it's possible that the virus could run out of people to infect, even at 0.5% of the population getting infected for 26 weeks or of the last 39 and 2.5% for 13 weeks that's only 45.5% so we'd need another 8 weeks at 3% to reach 70%. However even in the very high cases recently it's only been about 2% and with plenty of areas lower than that it's clear that case numbers could have carried on upwards for some time, most likely pushing it to a point where the NHS was failing.
As they had already put in place measures like using medical students and 1:4 treatment on ICU rather than 1:1, as such it's unlikely that it could have continued with much more capacity being needed.
Okay, but that doesn't answer the question, why is a vaccine the only way out of this, as you so boldly asserted?
As I said before, what are the other ways out?
As things stand a vaccine is our best hope, without it we would be looking at restrictions continuing for a long time. With it once cases are reduced and we've got a decent number of those at high risk of needing to go to hospital vaccinated and an ongoing programme of vaccinations then there's scope to start to remove restrictions as the more limited scale of infection should be manageable. However if there's ongoing concerns of importing a new strain that isn't limited by the vaccines then international travel could have restrictions for some time.