There is no reason not to fill out the FAFSA. Even if you have a lot of assets and high income, your child will generally qualify for unsubsidized federal loans. It won't be enough to cover school, but I don't know why you would stick it to your kid and refuse to furnish your information so that they can at least explore their options. Also, it is probably required in order for their universities of choice to put together a financial aid package. At no time does filling out the FAFSA mean that you have to cosign.
But without filling out the FAFSA, you deprive your child of the information needed for them to even formulate a financial plan without you. They need to know how much they're coming up short in order to save that money. From an 18 year old kid's perspective, your refusal sounds like, "Eh, I don't care. I didn't bother saving anything and now I can't be bothered to fill out some form. I don't care if you go to college. Not only am I unwilling to help you financially, I am unwilling to spare a modicum of effort for you."
Not filling out the FAFSA is like refusing to go to the doctor even though you think you have cancer. You're going to die anyway, right? Wrong. You don't really know what the FAFSA is going to churn out - and neither does your kid, until you fill the darn thing out.
As for the OP's original post: The military is an option, but any 4-year stint in the workforce can probably get you there. For example, if a child works and takes community college college classes half time while living at home. Save the salary, pick up whatever unsubsidized loans they can. Maybe the parents can scrape together $100/month for college savings during that time. Depending on the job, after 4 years they could be in a position to go to university full or part time (only 2 years behind, because they've taken the equivalent of 2 years' worth of community college classes).
The military has some obvious advantages over any other occupation, but it has some clear disadvantages too. For some people, free college just won't be worth it. For others, it's a no-brainer.
If the child has great grades, take the shotgun approach to university applications and scholarships applications. It is possible to get a full ride. Not all scholarships are need-based.
The most important thing is to let your child know early on that there won't be a lot of financial aid. My mom told me young that there would be no government aid (she was incorrect - she misunderstood the Canadian version of the FAFSA) and there was no college fund. So I worked in high school and saved money and I worked hard to get good grades. I ended up with a full-ride. If I had not gotten a scholarship, I would have continued working - since I had a job in high school, I already had my foot in the door for a transition to full-time. I have no hard feelings about my parents not being willing to cosign anything and not saving money because it was explained early (and often - since the message doesn't always sink in when you tell a 13 year old something) and I had a lot of time to prepare.
EDITED TO ADD: When I was in high school, the culture of the school definitely pressed people into immediately going to college. There was a feeling that everyone went to college right away and if you didn't, then you were a loser. On the other hand, I worked at a grocery store. I was surrounded by people who were out of high school and working to save for college/going part time. I think it helped with the bitterness to know that there was a different kind of normal out there. If the only people I knew were the kids who said, "Why don't your parents just co-sign?" then I think it would be hard not to be bitter.