> the added energy will definitely also make men more willing to do the hunt

Perhaps it is you with your interpretation that is misogynistic. The person was writing focusing on men, but both men and women are on the hunt for a compatible partner with which they could build a fulfilling relationship.

You called women trophies, they didn't. You can hunt for a job etc, it is a general term for when you put in effort towards finding/getting something.

This feels like a rather uncharitable interpretation. Entering into in an exclusive relationship with someone means you have come out ahead of other potential "competitors" and are able to claim the reward: access to the elevated relationship with the said individual, and not necessarily the individual themselves.

Do you also take issue with the phrases like "How to win his heart" or "You need to win her over"?

It's the reality of biology. You cannot avoid this.