If there is any contribution that I can make, let me add a clarification first, and then answer your question directly. There are a lot of confusion about the topic (frequestist properties of bayesian procedure), and even disagreement among specialists. The first misconception is "Bayesian intervals are meant to contain the true value of the parameter with probability $p$." If you are pure Bayesian (one that does not adopt any frequentist notion to evaluate the Bayesian procedure), there is no such thing as "true parameter". The main quantities of interested that are fixed parameters in the frequentist world are random variables in the Bayesian world . As a Bayesian, you do not recover the true value of the parameters, but the distribution of the "parameters", or their moments.
Now, to answer your question: no, it does not imply any assessement of the Bayesian method. Skiping the nuances and focusing in estimation procedure to keep it simple: the frequentism in statistics is that idea of estimating an unknown fixed quantity, or testing a hypothesis, and evaluating such procedure against a hypothetical repetition of it. You can adopt many criteria to evaluate a procedure. What makes it a frequentist criterium is that one cares about what happen if one adopts the same procedure over and over again. If you do so, you care about the frequentist properties. In other words: "what are the frequentist properties?" means "what happens if we repeat the procedure over and over?" Now, what makes such frequentist properties good is another layer of criteria. The most common frequentist properties that are considered good properties are consistency (in an estimation, if you keep sampling the estimator will converge to the fixed value you are estimating), efficiency (if you keep sampling, the variance of the estimator will go to zero, so you will be more and more accurate), coverage probability (in many repetitions of the procedure, a 95% confidence interval will contain the true value 95% of the time). The first two are called large sample properties, the third is the Neyman-genuinely frequentist property in the sense that it does not need to use asymptotic results necessarily. So, in sum, in the frequentist framework, there is a true and unknown value. You estimate it and you are always (except in a rare lucky accident) wrong in the estimation, but you are trying to save yourself by requiring that at least under a hypothetically indefinitely repetition of your estimation, you would be less and less wrong or you know you would be right a certain amount of times. I won't discuss if it makes sense or not, or the additional assumptions required to justify it, given it was not your questions. Conceptually, that is what frequentist properties refer to, and which good means in general in such context.
I will close by pointing you this paper, so that you judge by yourself if it makes sense and what it means a Bayesian procedure to have good frequentist properties (you will find more references there):
- Little, R., & others, (2011). Calibrated bayes, for statistics in general, and missing data in particular. Statistical Science, 26(2),162–174.