What you write is true, but these facts should be seen as imposing practical limitations on science. Sometimes, scientific inquiry will stumble onto ideologically charged questions, and the less aware the scientists are of the ideological implications, the greater the chance that their work will be sound. If the ideological implications are clear, the partisan opinions impassioned, and the consequences for practical power politics undeniable, we can't realistically expect that the results will not be influenced by these considerations, whether consciously or not. And if the scientific work is specifically motivated by the fact that the question is interesting for reasons of ideology or policy, the confidence we can have in its quality is very low indeed.

For all practical purposes, this imposes limitations on the efficacy of institutional science of the sort we have today, and this must be recognized by anyone whose interest is finding truth rather than ideological ammunition. There are already many research areas where the ideological influences are so strong that their output can be trusted only after a very careful examination, and there are those whose output is almost pure bullshit, yet nevertheless gets to be adorned with the most prestigious academic affiliations. Therefore, it seems pretty clear to me that in the present situation, science is already excessively engaged in ideologically sensitive areas, and encouraging further such engagement will result only in additional corruption of science, not bringing clarity and rigor to the discussions of these areas.

Take your example of vaccination. In a situation where researchers consider it a moral imperative to dispel the crackpot conspiracy theories and pseudoscientific claims about vaccination, I have very little confidence that their research will provide an accurate picture of the risks and negative consequences of vaccination if their magnitude actually is non-negligible, for fear of providing ammunition to the anti-vaccination side. Now, in this case, it does seem like the situation is simple enough that all evidence overwhelmingly points to the pro-vaccination side, and assuming agreement on the facts, there is no significant additional disagreement on values and preferences, so there isn't much concern overall. But often neither is the case, and insisting that science should be involved in the controversies more heavily will ultimately just corrupt and debase science, not bring any clarity to the situation.