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No, you shouldn't. It's obvious that any job that "requires" this, will not be a job that you want to be in.

I do all my work on private repos. My github looks bare. If they want code I'll send them code. If they want my github profile, I'll send them that and tell them all my code is private, hit me up for it/examples.

no you're fine without a filled github account. most engineers don't write code during their free time and, like you, work in private repos for their corporate job.

What is "grind mentality" in this context? Is this common in the programming world too?

At least from my (single data point) perspective it is. I feel the pressure to always be developing the next greatest thing in the next greatest language or framework and producing conference/journal-worthy research papers (I work in research).