Which is why the authors shouldn't say things like "the warmer they felt toward women, the warmer they also felt toward men… contradicting any notion that feminists’ ingroup love for women translates to outgroup hate for men" which you quoted! The first part is data. The second part is interpretation of "how it works". You have to do the proper controls and/or modeling in order to actually know.

For instance, suppose that really lovey-dovey people are more likely to be feminists. The feminists who aren't lovey-dovey are trained to hate men, and do hate men. The feminists who are lovey-dovey are also trained to hate men and they like men less than they would, but they like everyone so much that they're still very positive. Overall, even though feminism uniformly has a man-hating influence, you find that "feminists" like men more than non-feminists. (This is an example of Simpson's paradox.) If this were true, the conclusion would be "feminism trains people to hate men but is not effective enough at it to overcome the selection effect of it appealing to loving people".

Alternatively, suppose feminism trains people to love women. But people overgeneralize, so some of that rubs off on loving men more, too. In this case, the conclusion would be "feminism trains people to love each other: especially to love women, but also to love men".

Or, suppose that what we call "feminism" is actually dichotomous. We use one word, but there are two very distinctly different patterns at play: one successfully trains people to hate men, and the other successfully trains people to love women (and also, to a lesser extent, men). If the latter group was bigger than the former historically, then the measured effect will be that feminism has caused people to love women a lot more and men somewhat more. But if the group sizes have switched--and the study would have little way to know--then fostering more of the extant feminism would cause people to hate men. In this case, the conclusion would be, "Wait a second, 'feminism' is too broad to even use to answer this question beyond 'it depends'."

On a second reading, I'm less worried about the occasional misinterpretation of statistical results. They do make mistakes (e.g. "there was no interaction effect (p = .077)" in the supplementary--not reaching statistical significance isn't the same as there being no interaction effect, just inadequate support for an interaction effect--and at other times they demonstrate that they know this is the wrong way to talk about it because they do a power analysis and give estimates of effect sizes that they ought to be able to detect, etc. etc.). But it's not such a big deal. Most of the cases are easily translated into a statistically sound statement.

The real problem is that your overstatement of their overstatements of the meaning of the findings start getting pretty far from what the data actually shows. The qualifiers are really important, especially given that Study 4 and Study 5 give statistically significantly different results for the same question (which is the main one you wanted to address). If you phrase the issue as a "how much" question then you're on much safer ground than if you phrase it as a "yes/no" issue.

This is a well-supported characterization of the results of the paper: "If you ask people questions about what they think about men, you find that on average, people who identify as feminist like men roughly as much as non-feminists do; there's not a big difference overall. But, oddly,nobody believes that other feminists like men as much as they actually do: feminists don't believe other feminists like men much, and non-feminists really don't believe that feminists like men much. Perception is way off from reality, here, and we're not really sure why."

These are not well-supported:

(1) Misandry isn't real.

(2) Feminist women don’t hate men any more than non-feminist women do.

(3) Anyone who’s spent any amount of time with feminists as a group knows that feminists don’t hate men.

These can all be fixed:

(1b) Misandry isn't common.

(2b) Feminist women don't hate men drastically more than non-feminist women do.

(3b) If you actually are a feminist, you're likely to have a more accurate view of how feminists actually perceive men than if you're not.

Simplifying is an important part of science communication. Conveying something that isn't what the science says is not "science communication", however. It's science miscommunication.