Anyone who thinks grants have anything more than a minimal role in nonprofit sustainability does not understand how noprofit businesses work, unless they are supported by a unit of government as an agent for the provision of human services, like Chicago Area Project which gets the bulk of its revenue from state grants.
Nonprofits generally earn the preponderance of their revenue on a continuing basis from donations by individuals and/or organizations/businesses. They work to develop large networks of interested donors by having a properly constituted board of directors- meaning that board members designated as 'money people', whose primary purpose on the board is to assist in fundraising, must meet annual donation requirements- either directly from that board member's pocket, or through the network of pockets that board member is able to access. The combination of a good set of 'money' board members, a savvy development director, events, charged services, grants, and systematic/consistently applied overhead costs all lead to sustainability. Schools and hospitals have an additional tool- they can actually earn the bulk of their revenue from investment income, which other nonprofits are not allowed to do.