First I must warn you both: i'm no expert on the subject.. yet
Ah! Follower, the song you are talking about was not composed by Jara, it was written by Violeta Parra. She was chilean and I like the song too, although I find it a little ironic since she is singing about gratitude but her voice is incredibly sad!
And about Allende/Pinochet. Well, unlike Mexico, when Chile got its independence from Spain, things went rather smoothly and it had the strongest democratic tradition in South America. In 1970 chileans elected Salvador Allende, who was a marxist and head of a coalition government. The problems started when he nationalized mines and banks (in Chile mines have a really important % in the economy; and in those times copper was one of the main exports)and began land reform.
The economy almost colapsed: national debt and inflation rose sharply, and foreign investors began to pull all their capital out. The US started putting pressure also. Then came General Augusto Pinochet and organized a military coup to overthrow Allende (1973). He did and in order to wipe out communism he took several measures: censorship and violence towards political opponents. And this is where all of the accusations regarding human rights violations come.
Anyways, the economic well being was restored, and the government returned most nationalized companies to private ownership (it still controls oil production I think). This strenghtening of the economy did a lot for Chile while we were in economic crisis after crisis (1980-2...) (oops in Mexico we get economic crisis every six years when a new president steps in, so I guess we aren't a good comparison point)
Then in 1988 Pinochet agreed to let chileans vote on whether he should remain in power. He lost and in 1990 a new president came, Aylwin. Pinochet still held a post in the senate that was supposed to be for life...
This is a very rough picture of what happened, and you know the rest of the story. I live in Mexico, and the last time I went to Chile was almost 2 years ago. The country is completely divided about the matter. And outside I am sure we get the impression that nobody in Chile wanted Pinochet and that his supporters are a minority group. No! you can't believe how many people there are that love him. So there in Chile it is really a thorny issue, because not everything of what he did was wrong and he didn't hurt everybody. Things aren't black or white...
And about Spain and the UK, one of the problems was that Pinochet helped Margaret Thatcher with all of the Falklands Islands war by allowing the UK to operate from chilean bases and use chilean infraestructure.
Hope I could help you guys. And feel free to ask although I don't guarantee I can answer everything!
Maybe we can find around here a chilean/chilean