Some states do not require a partnership to have a written agreement. Also, as a matter of black letter law, any partner can bind the partnership.
Both points are irrelevant here. Banks make their own decisions as to what they require in terms of documentation. You may require a copy of a partnership agreement as a condition of opening an account. You had good business reasons before. Now, with regulatory emphasis on identifying beneficial owners, you have even more reason to do so. Your customer's alternative is to go elsewhere.
Moreover, accepting the directions of one partner out of three or one partner out of 15 is decidedly foolish. If the partnership agreement you receive does not name a managing partner, your bank may require all partners to sign any relevant resolution.
Don't take advice from customers.