For science, the right question is not really to ask when "the patron system" died off, but how scientific funding arrangements varied over time. There was never any "patron system" (which implies it was somehow formalized), there are just different arrangements by which people who were not self-funded got funding/support. In China, for example, scientific work was long part of the larger bureaucratic work of the state, which was a patron of sorts. In Europe, at different times and places there were very different modes of funding scientific work. Galileo deliberately worked to get himself embedded into the House of Medici, to be their "in house" scientist to bring them glory and occasionally be useful. Isaac Newton by contrast did his work in the context of a university, aided at times by the capabilities of the Royal Society of London, which as its name suggests was funded and chartered by the King. Charles Darwin paid his way through inherited wealth, but Thomas Henry Huxley originally made his wages by giving public lectures with charged admission. The German chemists and physicists of the late 19th and early 20th century made their money through affiliations with industry. And so on.

The way I think I would answer your question, though, is that science professionalized in the 19th century. That is, its jobs and funding sources became much more regularized, and fell into a few specific categories, e.g., industry, government support, philanthropy, and education (the rise of the research university, which used undergraduate tuition to pay for faculty research time, happened during this period). The relative "weight" of those categories has fluctuated over time; in the mid-20th century, government funding became paramount. But even in the case of US R&D in the late-20th, early-21st centuries, these trends have not been fixed. The funding situation changes in important ways over time, and this does shape the kind of research being done.

Does individual patronage play a role? As a percentage of work, certainly less than it used to — there still are some cases of wealthy individuals who will pay for work to be done, or give money to foundations to dish out. But they don't make up a significant proportion of the work being done. When did that change? I'm not sure it ever made up as much of a fraction as people sometimes think it did, but certainly by the 19th century that was seen as a more eccentric way of doing things, and by the 20th century, while there were still a few well-known individual patrons (e.g. Alfred Loomis), they were very unusual.

(I don't know anything about arts funding.)

Could you please define patron system in this context?