> capitalism would say an individual has the right to own an idea
"Capitalism" says no such thing. Property under capitalism is a consequence of scarcity; property rights exist specifically because a particular good cannot be expended for two different purposes at the same time. As such, someone must decide how the property will be used, and the entity with the right to make that decision is designated the owner. Ideas lack scarcity—the same idea can be used for any number of purposes simultaneously without being used up—and consequently are outside the capitalist definition of property.
Assigning fake "ownership" privileges to ideas infringes on the exercise of actual property rights, effectively seizing private property (including capital) for public use to to implement a policy of subsidizing the creation of new copyrightable works or patentable inventions, which is an inherently socialist exercise of government power.