Its a hard go to get a LFS to donate any mature media as they will usually just go crazy telling you that "bacteria in a bottle" will work or that you don't need to be doing any of it anyway. Its extremely rare for any of them to admit that fishless cycling might be needed, its just so very far from what they think is a sustainable business model, they can't swallow it.
But that's no reason not to try being polite and trying! You might just luck into that rare LFS!
You are quite correct that mature media is really the only reliable way to speed things up, otherwise mother nature is pretty much guaranteed to deal out a 3 week to 2 month session of growing bacteria and you can bet the real time it takes will be pretty random! Might be fast, might be slow and there might not be much to look at to figure out why!
But the thing I found that was unexpected was that all that time was quite useful to me as a beginner: It takes a long time to get a handle on all the issues around a stocking plan, they are more complicated often than one thinks. It also surprised me how much help it was to get some practice at tank maintainance.. water changes, algae and gravel cleanings during the second phase of fishless. Not having fish in there made it a lot easier for me to get the hang of it.
~~waterdrop~~