In this scenario, I am either scraping by on survival or putting all my efforts to escaping this Island. I am not built socially to survive with only my family, I'd go insane.
On another note, theoretically if I wasn't losing my gourd (big if). Yes I would still run, though running with people and racing makes it much more enjoyable. I don't really give a rat's ass who knows what my training or times are, but I like racing and running with people. I like competition, its fun.
More realistically, lets say we lived in an era prior to accurate timekeeping and social media. Which I think is more what you mean (or what you are after here). If running was a thing society could sustain, then yeah I'd be doing it. Would I be as fast training barefoot or in sandals/socks. No. But pretty confident I'd still slap a sub 5min mi easily, like I'd still run and race. Though we might not realize what the time actually is I guess.
Running helps regulate my eating and sleeping. I'm sure other exercise could fill that role, but I enjoy running for a lot of reasons. It's fun mostly.