It's not so much capitalism that breeds innovation, but competition. If there's no competition, where's the incentive for companies to innovate or improve in any way?

Take the private water companies in the UK as an example. You can't choose a different provider, and the service they provide has been getting steadily worse over the years, to the point where they don't improve infrastructure and we end up with Victorian pipelines pumping sewage into our rivers and seas. But nobody can change provider unless you move to a different part of the country, so there's no incentive for the companies to improve anything.