I'm on year eight of consistently chopping my willows to the ground 3 times a year or so, with zero apparent trouble for the willow. Thermodynamics be damned, willows are full of necromantic energy, and when killed - they rise again.
Cottonwood is taxonomically a close willow adjacent. I wouldn't anticipate it dying.
That said - cottonwood is amazing medicine, great fire board, fabulous carving, lovely baskets, and decent firewood. It's a tree I cultivate on my homestead.