What is bothering me, however, is that the founders believe the current direction is a dead-end, and keep encouraging people to come up with new product ideas. They are not involved in the process, except for organizing meetings where they take notes when we tell them something that sounds promising. There is no team work, everyone is kind of left to their own devices. Half of the team is not doing any work, actually, and they seem completely demoralized. The only difference between working in this place and not working is the (below average) salary. My analysis of the situation is that the company is functioning like a startup incubator: founders had no idea what they were getting into and are asking employees to build a company for them, except we get no equity.
Now, during my stay here I've come up with an idea and spent my free time working on a prototype that looks promising. Sharing my idea with the founder feels wrong. Should I leave and go my own way?
Be careful to clearly separate your work for the company from the work you're doing on the prototype. You mentioned working on it only in your free time (and outside of the company's offices, right?) but make sure to also not use any of the companies hardware you might have been provided. Depending on where you live and the exact contract you signed, the company might try to claim ownership of the intellectual property you've developed while employed even if it's your own thing.
The founders have clearly failed, you have a vision now. Make sure you can carry out the idea and direction and that your idea is commercially viable to avoid the situation that these founders have 'found' themselves in.
Good luck!