We model the world and the world follows the model and the exceptions add to our refinement of the model and that’s what makes it true.
Perception and reality are yin and yang and that is about as deep as my philosophy needs to go here
Although it can be challenging, this exploration can also bring greater depth to our perception and reality, much like the yin and yang you mentioned. Even if we don't adopt a new philosophy wholeheartedly, engaging with it can open us up to valuable insights.
For example the toplevel article says things like "independent existence in an objective reality".. this sounds important, right? Perhaps depending on the truth of the intuitionism, we might discover some facts about real world?
Nope. The whole "is intuitionism true" question only affects the person thinking about it. The whole books are written that basically quibble about dictionary definitions of "fundamental principles", "objective reality", "mental activity" and other complex words.
(This is especially visible in discussion of "consciousness": there are tons of texts about it, and yet none of them matter in any practical way. The practical applications -- generative text models, NLP, neuroscience, etc.. -- just ignore all the philosophic cruft)
It's now widely known that an intuitionistic proof has computational content (through the curry-howard correspondence)
Those things have a real world impact
For example, Robert Goddard, who invented first liquid-fueled rocket, became interested in space when he read Wells' "The War of the Worlds". So did Wells' work had an impact on rocket science? Certainly. Does it mean you should read "The War of the Worlds" if you want to build rockets? Probably not.
It's like knowing design patterns in OO: it's ultimately all software right? But software exists first in the mind of the programmer, and what's written in the IDE is just a shadow of what the programmer imagined. This makes one appreciate the point that there are things in software that exists entirely inside our heads
Studying history will very quickly dissuade you from this notion, over and over again. For the perhaps most striking example, consider how different the 20th century would have looked like if a certain philosopher named Marx hadn't tried to marry the works of another philosopher, Hegel, with economic theory.
(It's an interesting thought experiment what would have happened if there were no Karl Marx. February Revolution would have happened anyway.. but would bolsheviks still exist? Would Lenin still do all the things he did anyway, or would he become a civil servant instead?)
And for the most striking example, consider how different the 20th century would have looked if Academy of Fine Arts Vienna would have accepted a certain applicant from Linz. Does this mean fine arts educational institutions are very important in the political life?
Sure, given the economic and political realities of Russia at that time, something might have happened even if Marx hadn't existed, but it might have looked very different.
Philosophy matters to the world because other people read philosophy and are inspired by it to change something about the world. The French Revolution, US independence, Latin American independence and so on were all undertaken in no small part by people who had read a lot and were trying to make some of their ideals reality.