Maybe there is no need to complicate things by thinking about nano tech and advanced technology in the jabs. Maybe this one is just to give as many people cancer as possible because that is a type of death which they can easily deny. Steadily increasing rates of cancer, "could be anything" they'll say.
Moreover, with Earth's weakening magnetic field during the Pole shift, that will theoretically bombard Earth with far more space radiation than people are used to. It could be a one two punch, weakening people's immune systems and then kill them off with cancer. Not everyone of course, but who knows, perhaps hundreds of millions or more.
Maybe the full roll out of the more advanced tech comes in response to this first wave of death. Who knows and hard to say, but just thought I'd speculate here.
I am inclined to agree with your theory.
No need for fancy tech especially when they needed billions of poison shots.
What was in them was probably some sort of poisonous substance. Not too serious otherwise they'd all drop dead or something. Just a little poison to shorten lifespan. Few unlucky ones reacted badly to it too soon. Maybe they had different concentration of the poisons in different batches. It's easy to create poisons of that sort.
Now it seems the whole vax thing wasn't the end goal. Just a step on the way. That said, I don't think all of their plans are succeeding as they hoped they would.
Some of the best moves in chess begin with an apparent blunder. In fencing you might call it a feint-disengage attack. In WWE wrestling the loser and the winner both get paid at the expense of the audience.