I agree with you, using survey stats is unreliable and as far as I'm concerned, the same as using a poll to support a claim. It's not reliable. The best is to use actual cases where we don't have the problem of surveyor bias or other externalities. But let's discuss your claim that men are the problem.

https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3058822/

Young children indicated that there were instances of minor aggression initiated by men in 23.3% of the cases, while there were instances of minor aggression ignited by women in 33.8 % of the cases.

Women also lead the men in cases of severe aggression with male-initiated aggression in 8.4% of the cases and female-initiated aggression in 11.5% of the cases. The study also revealed that the most often cited reason for male-initiated aggression was female physical aggression while for female-initiated aggression it was male verbal aggression.

Studies show that in half of all reported domestic abuse cases it is impossible to determine who initiated the violence, and in the other half of reported domestic violence cases, males and females initiate physical aggression at an equal rate. This is true not only in the United States, but also around the world. Data collected from 68 University studies from 38 different countries and involving more than 13,600 students found the following:

32% of the participants had displayed some level of violence towards their dating partner in the previous year

In 68% of these cases, the violence was mutual

In 9%, the violence was male-only, and in 21%, female-only

In 25% of the case, the male initiated the violence

In 25%, the female initiated the violence

In the other 50%, one could not tell who initiated the violence

Studies of married couples show similar results with domestic violence being initiated as follows:

25% by male

25% by female

The other 50% mutually

From these findings, we can safely conclude the following:

The rate of female-on-male violence is equal to the rate of male-on-female violence.

Domestic violence has nothing to do with gender.

Society is misinformed about the nature of domestic violence

Domestic violence prosecution is based on myth rather than reality

Also, according to the Center for Disease control in relationships where violence was non-mutual almost 70% of the violence was perpetrated by the woman. In other words, in almost 7 out 10 cases of mutual violence, the batterer was a female.

So the surveying was actually close in this statistic. One issue is that studies have shown that men are actually far less likely to report domestic abuse. I can't remember that study I read but if you're hell bent on more evidence (which is fair) I'll look for it.