It's helpful to have an interactive browser when dealing with scRNA-seq data. There are many options if you want to do it on your own machine, but the choices are more limited if you want the share the data with a wider community. There are options for storing and sharing raw sequencing data, such as SRA or ENA. There are options for generic data, such as Zenodo or figshare. Why is there still nothing similar for single-cell data?
Many people set up a custom browser (many examples here), so there is definitely an interest in this type of a resource. If you want to go that route, you have to be responsible for setting up and maintaining some kind of server, which does not seem like a reliable long-term plan (just try to access a URL in a publication that's a few years old). There are two scRNA-seq specific resources that I am aware of, Broad Single Cell Portal and UCSC Cell Browser, but it's not clear if these are meant as a permanent resource and both feel like they are in still in beta mode (for example, UCSC data submission is a manual email-based process). Are these really the only options?
Single cell expression atlas from EMBL-EBI.
As for your question about a public browser, the culprit is likely lack of funding in short term. Plus a need for proper curation/management, which means real human involvement. This would add to the prior need on top of the hardware resources.