^^^ The above is correct. Like when Amba - in her original life, after the defeat of Parashurama - searched several kingdoms for a warrior who would cause the death of Bheeshma, Drupada
rejected her pleas. She had a garland whose wearer would be the cause of Bheeshma's death. After Drupada rejected her request, she hung it on the gates of his palace, and when she was reborn as Shikhandini, she recalled what she had done as Amba, and wore it herself.
Had Drupada had any enmity w/ the Kurus, he'd have accepted Amba's request, which he didn't. His enmity was only w/ Drona and that to after his defeat by Arjun. However, after Draupadi's vastraharan, all Kauravas got to be hated by the entire Panchala kul, irrespective of personal agendas.
Only thing I wonder - if Drupada wanted someone other than Arjun to win the contest, why design it for a perfect archer? Something involving a mace would have been easy for Duryodhan to win, if that was his goal. Or else, why even give Draupadi a swayamvara - invite Jarasandha's Sahadev over and hand her over to him.