I think science includes the type of studying that allows people to create good websites and computer software and hardware, and good music and movies and so on

Thats so incredibly broad as to be a useless definition of "scientist." Lets use "scientist" as someone engaged in basic research oriented around the natural word (as opposed to an engineer involved in more applied research). My categorization isn't perfect, but your grouping puts musicians,actors, programmers, actuaries,engineers,etc all in to an umbrella category of "scientist."

Most fundamental research happens at public institutions under public grants (even the private institutions get massive public subsidy). Also, as a matter of public-goods, economic theory would expect private institutions to be systematically underinvested in basic research.