The secular service has to have a clearly stated objective, that objective must be measurable, and that objective must also be something that relieves the burden of the State to care for people.
I do not believe this is true. Universities and museums rarely have measurable objectives, often don't even have clear objectives, and are usually 501c3. Lobbying and advocacy groups like the Sierra Club are 501c3, despite not relieving the state of a burden.
Article: Can investors make money in social services?
According to the article, six states are experimenting with funding social programs using "social impact bonds", which only pay out if the programs achieve their official objectives. The project is in its early stages, so it's not clear what will happen if the market decides that a program isn't worth funding, but this looks quite promising.