What problems are left for starting a business? I see only solutions searching for problems. Maybe come up with a competing solution to an already solved problem?

As a technology startup your root goal would be an exit. So you would work on problem which is in the roadmap for a big company and your wish is, they are not agile enough to build it themselves quickly and would buy out your stuff to beat their competition. So it's kind of contract work, with contract coming after the work is done.