I think it's pretty obviously a big part of it. For me, though, I've found that it's more important to examine the "wants," and determine what's actually going to add value to my life. Fancy coffee is nice and is one of my guilty pleasures, but in the long run, it's not going to make me happier to stop at the coffee shop every day. Being able to spend time outdoors, and to pursue fitness goals (I'm a competitive powerlifter), both add a lot of real satisfaction and value to my life.
So, it's worth it to me to give up my coffee habit, and put that money towards lifting gear and meet expenses. Likewise, one of my motivations for achieving FIRE is to have the time and resources to take 6 months to thru-hike the Appalachian Trail, and my big dream is to eventually move out in the country and set up a small farm/homestead type of thing.
Keeping my eyes on the big things puts my decisions in perspective. I have a lot of little wants, but when I think about the big wants, it's easy to give up the new shoes, makeup, coffee, etc.
There has to be some room for "wants" in the budget; if you feel deprived or like you can't enjoy anything, you won't be successful at being frugal. The key is to figure out what non-necessities add value and meaning to your life, and which don't, and living accordingly.