I also think part of where the issue comes from is the way that women communicate what they want to men. Women are every bit as bad at communicating as men. Men hear women saying "looks aren't that important to women" while missing just how much heavy lifting the word 'that' is doing in that sentence. Women don't want to be seen in public with an ugly man, they don't want to be made fun of by their friends and relatives for dating a man who they have to support financially, and what they're really communicating with the above statement is:
"women, like men need a baseline level of attraction. Women care about looks too, but aren't willing to put up with someone who only brings good looks to the table. Women need more than good looks. Appearance alone isn't enough to sustain a long term relationship"
I don't think they want to say it that way because of a fear of coming off as greedy and wanting it all, so they'd rather leave that extra part out and just hope that the man will figure out what it is they mean when they say that looks aren't important. Men need to remember that from the moment you've had your first day at school, your life will constantly be a competition. You'll compete against other kids in school so you don't fall behind academically and to get into a good college. You'll compete for the same job as thousands of other applicants when you're able to work. You'll compete in the work place to stay in the game and not get fired, and then again to climb up the corporate ladder.
If you're competing in all these avenues, why would you ever think you're not also competing in dating? It's plenty of men out here vying for the same women you want, and many of them are bringing more with them than you have. You can't kick back and just live your life and have a great woman walk into your life. That's a luxury that is reserved for women.. You need to put yourself out there, risk rejection, learn from that, and keep trying again until you eventually get someone whilst recognizing its literally thousands of men doing the same right now and all aiming for the same pool of women.
If having to compete all the time is stressful, then you can decide to focus your efforts on work, and making yourself as happy and healthy as possible. Nobody has ever died because a woman didn't fukk with them.