I would broadly agree with that analysis.
We currently have an average of 4,100 cases per day. The last time we were at that level (pre-vaccine rollout) was 22nd September last year, when the weekly and fortnightly % increases were also similar to today (i.e. we were on a similar upward curve).
At that time, we were averaging 23 deaths per day compared with 7.8 deaths per day at the moment, so we are now experiencing roughly one-third of the Case Fatality Rate that we were in the autumn. If you apply that to the 1,200 deaths per day at the peak in January, that would equate to approx. 400 per day instead.
Now, that's probably still higher than many people may be expecting given the efficacy of the vaccine. However, there are good reasons to believe that the third wave won't reach the same levels as the second wave.
Firstly, as you say, more and more people are receiving first and second jabs every day. The most recent deaths will have resulted from the situation as it was at least three weeks ago, and we have delivered another 3.6m first jabs and 7.1m second jabs since then.
And secondly, back in September we were moving towards winter with its propensity for respiratory illnesses and people spending more time indoors. Now, of course, we are heading towards the summer months which will, hopefully, be decent enough to encourage people to spend more time outdoors, where it is much harder to be infected with the virus.
Still a need for us to be cautious in our further relaxation of measures though.