Conditional probabilities and imperfect vaccines

Why is it plausible most future Covid-19 hospitalisations are among those vaccinated?

In recent reporting, BBC journalist Marianna Spring showed a sticker left by protesters:

A sticker reads:

“60–70% of hospital admissions and deaths are from people who have had 2 doses of the vaccines.”

That sticker provides a short link to “NHS SOURCE”. The document is not, in fact, from the National Health Service.

It is from the modelling sub-group advising the UK government. Despite the exact quote-marks, those words do not appear in the document. The sticker intends to paraphrase this part:

32. The resurgence in both hospitalisations and deaths is dominated by those that have received two doses of the vaccine, comprising around 60% and 70% of the wave respectively. This can be attributed to the high levels of uptake in the most at-risk age groups, such that immunisation failures account for more serious illness than unvaccinated individuals.

Given the nature of the protest, the implication is vaccines are thus ineffective. The figures do not refer to current admissions…