As a matter of historical fact, the Gracchi did harm to the Republic. While it is absolutely true that the Senate crossed a line in beating one to death and killing the other in street wars, the Gracchi had moved so far outside the traditional parameters of political protocols that their opponents could (whether justifiably or not) claim that they now represented a threat to the very fabric and ideals of the Republic and, as potential tyrants, had to be removed. So removed they were. Rather definitively.

I would argue that three factors interacted in this tragic episode.

First, the fact that the Republic was governed more by tradition than constitutional law. There were some laws of a constitutional nature (establishing the cursus honorum, for instance, or limiting the imperium of proconsuls), but most business got to done on the basis of established precedents, patterns of behavior sanctioned by tradition. Since there was so much redundancy in the system, this meant it was unclear where the boundaries of authority lay between, say, the consuls and the tribunes, or the senate and the tribal assembly of the plebs.

Second, the competitive and popular nature of aristocratic Roman politics left the door open for ambitious types -- whether well-meaning or not -- to take up a popular cause that divided the normally well-aligned people and aristocrats and then use said constitutional uncertainties to push their agenda. This Ti. Gracchus did, followed by Caius ten years later. I find it inconceivable that the Gracchi were entirely blind to the personal political benefits to themselves and their prominent family had their agenda passed into law. This does not mean that ambition was their only, or even their dominant motivation, but it MUST have played a role in their political calculations.

Third, and finally, since the Republic held to a reactionary anti-monarchic ideology, and since the established way to treat a tyrant since Greek times was to kill him, when the Gracchi took advantage of factor 1 to enact factor 2, they could be portrayed by their political opponents as aspiring tyrants out to overthrow aristocratic privilege, and so had to be justifiably killed. This is the aristocracy's take on the matter, as evidence by Cicero's comments on them.

The results of these interacting factors were the tragedies of 133 and 121 BC. And, since the system worked on precedent, the new precedents set by the Gracchi and their opponents alike spelled disaster for the Republic.