I'm not an expert in the medical field, but I think the maths works a little differently here.
If a vaccine immunises you from the disease, then even if you get infected, you're not going to be suffering from it and infecting others for long, if at all. (Further research on the degree of this is underway, so said Radio 4 this week.)
The main way a vaccine helps the county is by reducing the spread of the disease. Do you remember the diagrams produced in the Spring, showing how if one person infects three others, then those three infect a further 9, they infect 27, and you end up with ~250 getting infected in the 5th round of spreading? And how social distancing reduces the number of people we might spread the disease to, and avoids that exponential increase?
Vaccines play the same game, but from the other end. Let's say (to make the numbers easy) that 75% of the population takes up the vaccine, which is 90% effective. 90% of 75% is roughly two thirds of the population that will have immunity to the disease. So if your day-to-day actions mean you might spread the disease to three other individuals, two of them will now be immune, and the numbers won't spiral any more.
That is what herd immunity is: a situation where enough of the population is immune (through prior exposure or vaccination) that things won't spiral out of control. And that's how we can afford to not vaccinate those who are clinically vulnerable to everything including a vaccine, or those that object on religious grounds, while still behaving normally. If people around them aren't spreading the disease, they ought to be safe.
If the disease isn't spreading any more, then even with a 1% fatality rate, you'll see far fewer than 0.1% of the population succumbing to it in the future. Most of that 1% who are fatally vulnerable will never encounter the disease.
The flu is as deadly as it is because we don't have an effective vaccine for it (because it mutates every year, and only a subset of the population ever gets vaccinated). While the jury's still out on how Covid mutates, we're definitely going to get far more of the population vaccinated, so it should become far less of a threat.
Taking your 2/3rds protected figure (which probably is a probable outcome, even if we don't get to the 75% take up due to people potentially taking other measures, like those at high risk but can't have the vaccine limiting the number of people that they interact with - not this doesn't mean locking themselves away just being careful about who they interact with) whist there'll be some who infect more than one person there'll also be those who don't infect anyone.
However with no other protections in place rather than it being 1,3, 9, 27, 81, 243, 729, 2,187, 6,561, 19,683, 59,049, etc. It shifts to being 1, 1, 1, 1, 1, 1, etc.
Now given that we're at 119,000 cases a week currently of we can get the R value to 0.75 the numbers start to fall away rather quickly from 119,000 to 89,250, 66,938, 50,203, 37,652, 28,239, 21,180, 15,185, 11,913. Therefore infections could fall to 10% of current values within 8 weeks if we could vaccinate 75% of the population within a week and had no other measures.
The problem is that it's going to take months to get there, however the good news is that there are currently other measures in place. As such case numbers should start to show signs of falling within a short timeframe (as even though you don't get 90% from day one, you do get some protection from within a few days of the first dose) due to all that's happening currently.
Therefore it should be possible to see many fewer areas in tier 3, and hopefully also tier 2, by Easter. Whilst that may not be normal enough for everyone is a good way towards it. With some other measures (test and dine/drink) we could get a lot closer to normal by that sort of timeframe.
There's likely to be some who like some of the measures in place (not masks, but rather things like more internet shopping, WFH, having an excellent excuse not to see the in laws, etc.) and so continue those measures even when they are not mandated which would help reduce the R value.
That’s the undying issue here I think. People are struggling (understandably) to separate the science from the politics. This can probably be attributed to the scientists standing alongside the politicians and delivering manipulated statistics and, frankly, misinformation live on TV. A significant number of people no longer trust the experts when it comes to Covid-19 and our response to it. Not only that, but I’ve been surprised at how many people I’ve spoken to who believe there is ‘something’ going on that we aren’t being told about. I don’t mean the crazy conspiracy stuff, but ‘something’ that isn’t necessarily in our best interests. They therefore want to push back in anyway possible, and not taking the vaccine is an obvious way to do so. I’ve had some very interesting and often surprising conversations with people over the last week!
Personally if I wanted to "engineer" the need for us to be "vaccinated" there's simpler ways of going about it.
In fact I'd use the fact that so many people have been avoiding getting their children vaccinated as the reason.
I'd find an area where there's been an increase in Measles and start an adult vaccination program based on the need to ensure high levels of uptake. If I was being mean I may even spread miss information about how there were some bad batches of the vaccine from the 80's and 90's, as well as blaming those immigrants who've come here without having the right vaccines. Furthermore I'd deliberately highlighting (through an over zealous media campaign) each and every death and other heath issue which I've said that it's caused. Obviously with the throw away line "this is only those which are know about, how many more are going unnoticed or intentionally hiding from us" (possibly even blaming immigrants further on that last point).
Then bring in a policy starting that you couldn't come into the UK without an updated vaccine (again blame immigration, maybe even create a bit of miss information about how the EU have stopped us from bringing in such a rule for to freedom of movement).
It would be much easier to manage and although may not create the same level of fear, would result in a significant number coming forwards for vaccination (especially if you wanted to go abroad for a holiday).
Then just microchip everyone. Happy days....
Of course on a more serious level, there's almost certainly lots of stuff that the government aren't telling us. Whether that's a reason not to be vaccinated is an entirely different matter.