I'd argue that the answer to that question is "yes", no matter if it's Blizzard or some other company. It's completely ethically valid to boycott a shitty company with shitty leadership and shitty culture (as OP puts it), however, it's also completely ethically valid to not boycott such a company and simply pick the best product. There is no ethical duty to participate in every fight; it's ethically good (and better) to "fight for good" but it's also completely ethically acceptable to not put effort in any one of the near-infinite of those fights - it's only unethical if you're actually harming others or doing evil, and just living your life is not that.
Guilt by distant association is an abhorrent concept that should not be validated - using someone's products does not imply supporting the company's policies or leaders/owners personal behavior or political actions. If you want to make a point of ostracising someone, sure, go right ahead, but there's a very high bar to pass to state that it would an ethical requirement for everyone else to participate in that ostracism - IMHO it might be debatable if it would be a an ethical requirement in case of clearly convicted murderers, and even more so in cases which are (a) less severe than murder and (b) the particular culprits less clearly proven.