To answer your question directly no, the bulk of on-demand models - as a stand alone offering - do not have the economics to survive because of the inability to define a true demand curve. Recently, Freakonomics Podcast explored this reality using UBER data and the challenges economists have had in this way.
Why might this be the case?
There might be too many "UBER" models in force. Many have hopped on board given the promise of lower costs of and commitment to human capital. Optimizing around cost savings only may distort value delivered to customers.
Discovery isn't aligned with the tools people use most. Recently, Google began "streaming apps" like UBER through the browser to reduce the pain of downloading an app for single usage and provide a smooth gateway if request from Google Maps. Others may exist that have the same offering, but create barriers to access it.
Local substitutes often exist. On demand services often underestimate people's local knowledge. Rather than using a service like YourFuzzy to develop a relationship with your vet, picking up the phone on a friend's recommendation may prove to be a better approach.
Good on-demand models work well when desire can be fulfilled in a familiar and repeatable fashion. If there isn't a sustained desire for whatever service is being provided OR the offering is mis-aligned with when and where People need it, the economics become difficult to justify and the dataset either not large enough or too variable to provide clarity or any real value.
That said, I think the challenge of sustainability here isn't necessarily one of supply chain as you stated, but one of the demand chain: how such a service is discovered by would be customers as well as how this maps to value delivery and internally supported by "independent contractors".
To illustrate this, if your demand is mis-aligned or too variable, it's simply not worth the Contractors time to engage, resulting in a broken promise to customers. UBER and the like work well because it's use case is such that nearly everyone can imagine needing at one time or another. Because of this, they don't have to worry much about supply of contractors meeting demand save something like a blizzard.
Michael Fisher stresses the importance of understanding the nature of demand in his paper "What's the Right Supply Chain for Your Product?", stating that demand "pull" models work well for innovative products/services d to demand, lead time and other uncertainty.
In my mind, achieving economic sustainability become a little more clear:
Tighten Your Reach - Too many OD services go national with whats otherwise a more local issue. Effectively reaching 90% of 100k people is preferable to 2% of 30 million.
Make the offering more specific - This is accomplished by refining what's offered or through identifying when the service is necessary.
Align with where they are most likely to discover - This is a little more challenging. Understanding your customer base and espousing their ideals for where and when to request your service will not only give them a better value, but provides you a more consistent data set and decision architecture.