Whilst this may well be true, the very same problem affects COVID death numbers: a death is reported as a COVID death if it occurs within about a month of a positive test result, regardless of whether symptoms are displayed at all or whether the person is already dying of something else. Thus using the same standard to classify vaccine-correlated deaths ensures at least that the numbers are comparable in this regard. You can see where this affects their results where they calculate a CFR using the globally reported death rates.
If they had used some other way to define a vaccine-related death, it would make the vaccines look artificially good, because you'd be comparing it against something made to look more deadly than it really is.
There is an additional problem with your counter-argument: it is anticipated and is discussed in the paper, which appears to counter it:
"One might argue that it is always difficult to ascertain causality in such reports. This is certainly true; however, the Dutch data, especially the fatal cases, were certified by medical specialists"
They also discuss your second point:
"In the Israeli field study, the observation period was six weeks, and in the U.S. regulatory studies between four to six weeks, a period commonly assumed to be sufficient to see a clinical effect of a vaccine, because it would also be the time frame within which someone who was infected initially would fall ill and perhaps die. Had the observation period been longer, the clinical effect size might have increased, i.e., the NNTV could have become lower and, consequently, the ratio of benefit to harm could have increased in favor of the vaccines. However, as noted above, there is also the possibility of side effects developing with some delay and influencing the risk–benefit ratio in the opposite direction"
Some of the counter-arguments on Twitter seem to criticize the paper for being bad and then make other dubious arguments of their own (it's frustrating, nearly all COVID research is like this). For example "Health Nerd" states:
"10/n If you assume that everyone who stays alive will get infected without a vaccine eventually - which is a fact"
That is not in any way a "fact". That is a model assumption based on epidemiological theory that has not yet yielded any usefully accurate predictions. The theory is so weak you could have made exactly the same arguments on exactly the same grounds for SARS-1 or MERS-CoV or really any infectious disease at all, yet the world is full of infectious diseases and people who have never caught them. Additionally it ignores the way viruses seem to become less dangerous over time as they evolve to optimize for propagation rather than killing their host, meaning even the very nature of what people eventually may get infected with can be very different. This is already visible with the discussion of how the delta variant has milder symptoms than the original.