It has definitely succeeded. For example, New Zealand uses this model, and we're actually in the middle of rolling out fiber to almost every house in the country.

Okay, so technically in New Zealand the last-mile infrastructure is owned by a utility company, not a non-profit. However, the utility companies are selected by the government and tightly regulated, so they act more like non-profits than typical companies. Also note that they aren't allowed to actually sell internet access to consumers; they only maintain the infrastructure.