Nah, they'd still be extinct: https://en.wikipedia.org/wiki/Passenger_pigeon#Hunting
After being opened up to the railroads, the town of Plattsburgh, New York, is estimated to have shipped 1.8 million pigeons to larger cities in 1851 alone at a price of 31 to 56 cents a dozen. By the late 19th century, the trade of passenger pigeons had become commercialized
Even if adjusted for inflation, 31 cents a dozen doesn't sound like a lot, but then market saturation happened and your prediction came to pass:
The price of a barrel full of pigeons dropped to below fifty cents, due to overstocked markets. Passenger pigeons were instead kept alive so their meat would be fresh when the birds were killed, and sold once their market value had increased again. Thousands of birds were kept in large pens, though the bad conditions led many to die from lack of food and water, and by fretting (gnawing) themselves; many rotted away before they could be sold.
Those who don't learn from the past are something something