fyi most dairy cows are kept beyond the age of 3. You can see on the World Dairy Expo YouTube channel that there are cows producing up to the age of 6.

Heifers reach puberty at about 12 months old, but some delay until up to 20 months old. After breeding they gestate for 283 days (almost 9 1/2 months), give birth to their calf and give milk for about 10 months. "Culling" a cow at the age of three would mean only getting 2 full lactation cycles of milk from them, which isn't super economical.

You're not too far off the mark, though: once any individual animal becomes less profitable than a new one could be, they get culled.