It doesn't matter if you have a billion dollars, you need a proper immigration reason to stay. You cannot just come here and live without a proper visa.
The COE is a formality to most visas, and it is NOT vague.
Application for Certificate of Eligibility
Don't want a job? You are limited to 2 options: student visa or cultural activities visa (unless you want to start your own business, in which case you need an investor/business manager visa, but that involves far more than you are willing to put out; see the bottom link within the link I just provided on COEs).
Don't want to attend any kind of language school? You're down to one option. And, even then, you usually have to prove you have already done some studying of that cultural activity (martial arts, pottery, flower arranging, swordmaking, etc.).
In cases where the person concerned intends to engage in the academic or artistic activities, or to engage in specific studies on Japanese culture or arts; the following materials are required.
Documents certifying the activity and its duration of the person concerned, and the outline of the organization where he or she intends to engage in the activity concerned.
Documents certifying the academic career, professional career and other career regarding the activity of the person concerned.
Documents certifying that the person concerned can defray all the expenses incurred during the stay in Japan.
In cases where the person concerned intends to learn Japanese culture or arts under the guidance of the expert.
Additional materials showing the career and the achievements of the expert, besides the documents mentioned above are required
Your avatar says you are from Taiwan, and that does have a working holiday visa option, but the idea is to be traveling, not living here with kids attending school. Look into it, but I have my doubts.
MOFA: The Working Holiday Programmes in Japan
I think your options just went to zero.
What is the reason you want to be here that long and drag the kids along? That is, why won't a few weeks on vacation do? (You get 90 days as a tourist.)
MOFA: Exemption of Visas [Guide to Japanese Visas]