The Civil Media Co. bills itself as a solution to some pressing problems in modern journalism, pitching a new cryptocurrency that would help news consumers support quality information and weed out bad actors.

So far, the effort isn’t going smoothly. Consumer demand for the new so-called token that the company is selling has been weak ahead of an Oct. 15 deadline, and major news organizations have rebuffed partnerships with the company. Meanwhile, a Civil co-founder said its business plans are fundamentally flawed.

“We...