(This does not qualify as answer, since I hardly use any references beside the paper linked by OP)
No contradiction between claim about efficieny of bivalent vaccine and diagram
Note that the claim about vaccine efficiency refers to the new bivalent vaccine, whereas the diagram refers to previous vaccinations.
The bivalent vaccine they study contains the original vaccine (against the Wuhan variant) as well as vaccine against omicron BA.4/BA.5. All previous vaccines were developed against the original Wuhan variant only.
These are also the variants dominant throughout the duration of the study.
Note also that the claim in the paper is actually narrower than what the part quoted in the question suggests:
The bivalent COVID-19 vaccine given to working-aged adults afforded modest protection
overall against COVID-19, while the virus strains dominant in the community were those represented in the vaccine.
It is not known if the vaccine will be equally effective when the strains circulating in the
community are not those represented in the vaccine.
Finding (moderate) vaccine efficiency against (symptomatic) covid with a vaccine adapted to the actually circulating variant does not contradict waning efficiency of vaccines that were developed against variants that are not in circulation any more
(Unfortunately, they don't compare people recently vaccinated with the original vaccine vs. the bivalent vaccine. Table 3 lumps together infections with vaccination. This does make sense biologically, but prevents us from seeing whether the bivalent vaccine is more efficient than the original one.)
Increased incidence and number of vaccinations and possible confounders
Summary: Here it is most important to keep in mind that correlation alone doesn't tell us about causation, and in particular not about the direction of causal effects. After all, people may get vaccinated in order to not need to fear (severe) covid when infected, i.e. to to replace more stressful measures to mitigate their personal risk of severe illness such as avoiding contacts/exposure.
First of all, the paper reports that similar findings have been published before.
The study (like many others) actually evaluates a surrogate for vaccine efficacy, which is called effectiveness based on observed case rates. Since it is observational, any confounder that changes exposure risk and correlates with vaccine uptake can mess up the conclusions. Thus, we can correctly say that they observe a correlation, but we cannot conclude too much about vaccine efficacy (which is basically vaccine efficiency in its everyday sense).
A bit puzzling to me is that the authors state
Those who chose to get the bivalent vaccine were also more likely to have lower risk-taking behavior with respect to COVID-19 [than those who did not take the bivalent vaccine]
since there is no indication in the methods section that they recorded any indicators of risk-taking behavior.
I'd like to dig a bit deeper into the discussion they give in the paper:
A simplistic explanation might be that those
who received more doses were more likely to be individuals at higher risk of COVID-19.
"risk of covid" refers to two distinct types of risk or rather, likelihoods/probabilities.
In particular some people have higher risk of exposure than others (E.g., in the study population the subgroup "clinical" occupation may have a further relevant subgroup of people working in covid wards)
In addition, "risk of covid" is used to refer to the probability of severe illness when exposed.
proportion of individuals may have fit this description. However, the majority of subjects in this study
were generally young individuals and all were eligible to have received at least 3 doses of vaccine by the
study start date, and which they had every opportunity to do.
Since age has a very strong relation to severity of covid, but no obvious relation to SARS-CoV2 exposure in health care working places, this refers to the 2nd meaning.
But the study does not measure and model severe covid but something close to "sufficient symptoms to think about getting sick days"
Therefore, those who received fewer than 3
doses (>45% of individuals in the study) were not those ineligible to receive the vaccine, but those who
chose not to follow the CDC’s recommendations on remaining updated with COVID-19 vaccination, and
one could reasonably expect these individuals to have been more likely to have exhibited higher risk-
taking behavior. Despite this, their risk of acquiring COVID-19 was lower than those who received a
larger number of prior vaccine doses
My take on the confounder situation is that there are a number of plausible potential factors (and interactions) which would be expected to act into opposite directions. Some of which one may be able to address from the raw study data, but the aggregated information they report does not allow us to conclude much here.
E.g. people with high-exposure (workplace or other) may be more likely to have taken more vaccines.
This is the simplistic explanation the study authors refer to, which they dismiss as small proportion without giving or citing evidence.
Here in Germany, the official vaccination recommendations roughly boil down to saying that for each infection, skip one dose. Recommendation is also to wait 6 months ("in exceptional individual cases, 4 months is acceptable") after an infection with the next vaccination dose.
From that perspective, skipping (some) vaccination doses according to the infections one had does not imply particularly high risk-taking behaviour.
And an interaction with high-exposure settings correlating to lower total number of vaccinations (because of infections; note that the study population selects towards people who have recovered from their previous infections)
The raw data of the study should allow to check at least for such a correlation, and a plot similar to figures 1 and 2 but grouping by total number of relevant exposure (i.e. vaccinations plus infections)