First off, the Olympics are (and--at least in the modern era--have basically always been) something of a political pissing contest. Maybe there's something to be said for MINIMIZING the amount of "political football," but not boycotting the Olympics isn't going to convert a politically-saturated event into pure fun and games.
Second, failing to object to the status quo (in this case, refraining from boycotting the Olympics) is itself a political act. Whether intentional or not, failing to fight problems in the status quo is effectively equivalent to tacitly endorsing them. Ignoring the human rights crises in Russia for the sake of the Olympics is not the "neutral" choice; it's every bit as political as the decision to boycott (in effect, at least; your intentions may be entirely apolitical, but the LGBTQ folks being jailed and beaten don't care much about your intentions on this one if the effect is the continuation of systems of discrimination).
Third, however, I'm not entirely certain that a boycott would solve or help to solve the problem permanently. While it might do so, I take issue with the common assumption that boycotts are always an effective strategy and that their benefits outweigh their costs (I also take issue with the extreme opposite assumption: that boycotts never work and that they're always too costly and risky). Even if any state was willing to threaten to boycott the upcoming Olympics (which I'm not sure is the case anyway), Putin would likely just institute a temporary moratorium on prosecution of LGBTQ folks. At most, he might promise to improve in the future. Neither of those responses are particularly likely to lead to meaningful long term change. There might be a way to approach a boycott that would fix these problems, but if so, it isn't readily apparent to me.
That said, if a boycott would likely result in meaningful change, the benefits of which outweigh the costs of boycotting (or threatening to boycott), AND a critical mass of states would actually be willing to participate in such a boycott as to make it effective, then sure, it would be a worthwhile approach to addressing Russia's human rights abuses.