SPA: using websockets only. Why not?

No, it won´t be a bad idea. Actually I work in an application that uses a WebSocket connection for all what is data interaction, the web server only handles requests for resources, views under different languages, dimensions .. etc..

The problem may be the lack of frameworks/tools based on a persistent connection. For many years most of frameworks, front and back end, have been designed and built around the request/response model. The approach shift may be no so easy to accept.

Coming back to this question a few years later, I would like to point out a few aspects to illustrate that having all your communication through websockets does have its drawbacks:

• there is no common support for compression. You can easily configure your webserver to compress http requests and browsers have been known to happily accept compressed responses for years, however for web sockets it is still not that easy (even though the situation has improved)
• client frameworks often are build upon commonly used standards like rest. The further away you move from frameworks expectations, the less addons or features will be available.
• caching in the browser is not as easy. By now this goes a long way, reaching into the realm of offline availability and PWAs.
• when using technology, that is only used by a subset of users, it is more likely to find new bugs, or bugs might take longer to fix. And if it's not bugs, there might be an edge case somewhere around the corner. This isn't an issue per se - but something to be aware of. If one runs into those things, they often easily take up quite some time to fix or work around.