近 walks 辶 with the axe 斤 lending sound: nearness measured on foot. What's 近 is what you can walk to — the ancient unit of 'close' was your own two feet.
最近 'most near' means 'recently' — nearness in time. 附近 is the neighborhood, 近视 is nearsightedness. The proverb 远亲不如近邻 says a close neighbor beats a distant relative — proximity is its own kinship.
Walking 辶 distance — NEAR enough to carry an axe 斤 over.
最近怎么样? 'how have you been recently?' is the standard catch-up opener — Chinese greets you through nearness in time rather than 'what's up'.