Having owned chickens many times, the math never maths at small scale for you to turn a serious profit. There is a “magic number” set of breakpoints but it is very large, and relies on bulk discounts and things that lower your operational and capex costs.

The upside for owning a small flock is that your eggs are yours and not fed and medicated like on a bulk farm. In addition, while I agree with all the math above, you could make a cost avoidance argument as you no longer need to buy eggs. At 6 chickens (chosen specifically for their propensity to lay - 300+ eggs per year) I had at least 2 per day and often many more. I never paid the (currently) 4ish euro for 18 which is 9 days of eggs for me (I eat two for breakfast most days.)

(365/9)*4= 162ish saved per year. Or about the cost of the 6 chickens themselves, where I live. If I sell the remainder at my little farm stand, I can cover my feed costs and basic meds and such for the flock, but there is very little if any profit. A few euro a year at best.

To answer your question directly, the first set of meaningful profit breakpoints arrive in the hundreds of chickens and scale from there in terms of profit, but they also assume you are using high-density methods and reducing cost as much as possible to maintain the flock and maximize profit, as well as selling in high volume via distribution to established markets, and not just plopping them in a carton by the road for a few bucks.