There is a local nonprofit that has been lying dormant for the past ten years. All the board members have died and no one has stepped up to get it operating. I went to the secretary of state here and filled out paperwork to become its registered agent. I've checked the bylaws and there's nothing written that prevents me from becoming the registered agent. On this basis, am I legally the nonprofit's registered agent since no one owns the nonprofit?
Probably not. The basic concept of agency is that someone is authorized to act on behalf of another. From what you say, no one with the organization has authorized your acting on its behalf. You cannot simply proclaim yourself an agent for someone who doesn’t authorize it.
But even if you were the registered agent, what would that get you? The function of a registered agent is to receive official notices, like tax notices or service of a complaint in litigation, and forward them to the officials of the organization. A registered agent has no right to run the organization, to appoint new directors to fill vacancies, or to take other substantive action with respect to the organization. Your entire job description would be to sit around waiting for something that is very unlikely to happen. And if it did, you couldn’t fulfill your duties as agent because there isn’t anyone to notify. I am not sure why you would want to volunteer to undertake an impossible job.
If you want to control a nonprofit, start your own. The old one may have been administratively dissolved. It has certainly lost any federal tax-exempt status that it might have had by failing to file a tax return for more than three years. There would be no reason to reopen this entity, even if you could. It would be much better to start with a clean slate.