Mumbai, Feb. 11 -- On most Saturdays, between the whir of drills and the shuffle of patient files in a busy suburban Mumbai dental clinic, you might find a 66-year-old entrepreneur bent over a whiteboard, talking unit economics instead of molars.

A. Velumani, the man who built Thyrocare into one of India's most efficient diagnostic chains and exited at close to a billion-dollar valuation, is back in builder mode. Not in diagnostics, but in dentistry. And not as a passive investor, but as an operating mentor with a stopwatch in hand and a disruption thesis in his pocket.

For the startup world, Velumani has lately become something of a folk hero-the scientist-turned-entrepreneur scaled with frugality, exited at scale, and now dispenses to...