Umbridge's failure to succeed in Slytherin boils down to this: Umbridge is an idiot.
Seriously. Yes, she's half-blood, but Voldemort, another half-blood, managed to become Head Boy while in Slytherin, and Snape, also half-blood, was part of the Slug Club. Umbridge would probably have attended school while Slughorn taught there, and Slughorn was one to recognize talent, even in Muggle-borns (Lily Evans Potter, for instance). Slughorn's only known comment on Umbridge, however, is extremely unfavorable: "Idiotic woman. Never liked her." It is unlikely that he would have passed over Umbridge for leadership positions, blood status or no, had she been truly qualified.
Umbridge's advanced magical abilities are evidently limited to being able to produce a Patronus and to develop a torture quill. Otherwise, she seems very incapable. Her Charms knowledge is, in general, very weak--she is the only Hogwarts professor incapable of removing Fred and George's swamp from the floor--as is her knowledge of magical creatures, or she would never have behaved as she did toward the centaurs. Her time in Slytherin does seem to have taught her how to recognize a fraud (Trelawney), but in general, she lacks magical ability.
Umbridge also does not seem to understand people very well, which is another reason she never rose through the ranks in Slytherin. She is ambitious enough to jockey her way up through the Ministry, but she does so in a very primitive fashion. She can only function in two modes--toadying to her superiors, and crushing her inferiors. That's hardly the mark of a master manipulator. Additionally, a more socially aware teacher probably could have toed the Ministry line without letting the entire class see through her immediately. Umbridge can manage as a Ministry bureaucrat, but as a teacher and as a school administrator, she only manages to incite everyone--even the other teachers!--to revolt.
While we never get to see Umbridge directly interacting with Slughorn, we do get to see her interact with Snape, a fellow Slytherin. Snape may not always get along with the other teachers--he disagrees with Dumbledore at times, and has a permanent Quidditch-related rivalry with McGonagall--but he seems to respect most of them. He does not respect Umbridge at all and hoodwinks her on multiple occasions. Snape's attitude toward Umbridge is probably a good indication of what other Slytherins' attitudes might have been when she was a student.
Umbridge's blood is not the problem. It's her lack of ability.