Dolores just didn't have a great start in life and it all went downhill from there. As we read on Wizarding World (formerly Pottermore).
You are correct, Dolores was a half-blood. A half-blood who resented both halves.
Dolores Jane Umbridge was the eldest child and only daughter of Orford Umbridge, a wizard, and Ellen Cracknell, a Muggle, who also had a ...
"The Elder Wand cannot serve me properly, Severus, because I am not
its true master. The Elder Wand belongs to the wizard who killed its
last owner. You killed Albus Dumbledore. While you live, Severus, the
Elder Wand cannot truly be mine."
"My Lord!" Snape protested, raising his wand.
"It cannot be any other way," said ...