I can give some background on the state of each location which should indicate the relevant push and pull factors. The condition of the kingdoms which will become Germany in 1871 should hopefully answer your question as to why the song is so negative towards their homeland.

Although the constitutional monarchy of Brazil would last until 1889, the early 1820s were a messy period for the country. The new Emperor Pedro I concerned many in court that he would attempt to rule as an absolutist under the influence of Portuguese merchants. Many native Brazilians were wary of Portuguese influence for it was Pedro's father ruling from Lisbon who agreed to set trade restrictions on Brazil which hampered its economic development. Although in the end, Pedro gave in to the creation of a constitutional monarchy, high levels of inflation and shortages angered many who expressed their anger at the local Portuguese. All in all, Brazil was in a bad state economically. Even though warm ties with Great Britain were been established (eventually resulting in the Anglo Brazilian Treaty of 1826), there was a requirement for Brazil to totally abolish slavery within three years of signing which would have further dragged down the economy.

In such cases of strife, it was not surprising for Brazil to look to immigrants to contribute to their fractured economy. Before Pedro's reign, the Portuguese were a natural fit but became politically and socially toxic (rioting was consistently directed at Portuguese immigrants in the next few decades). Politically, the new government saw divides within its society. Two thirds of the population were made up of mulattos and blacks (with a quarter of them been slaves) who were most likely to engage in rebellions when economic circumstances became too tough. With a racist tinge, this encouraged a desire to increase the white presence within the country. To act as a carrot, the government was willing to sponsor immigrants from Europe by providing transport to anywhere within Brazil upon their arrival and provide a hearty loan to be only paid back after the first harvest.

With this inviting offer, many in Spain, Italy and Germany were open to the idea. Europe post Napoleon was in dire economic straits. Proportionally, the same if not more people died during the Napoleonic Wars as they did during The First World War. Mobilisation had left farms and harvests unattended causing huge food shortages and near famine. Not just the French armies but Austrian and Russian had lived off the land for years in German territory stripping its assets bare. To make things worse, a volcanic eruption in Indonesia had ejected so many particles into the air above Europe the sun's rays were weaker than normal reducing in poor agricultural bounties. Early frost in Baden devastated the economy, the Rhine flooded for five months and a diary made in Stuttgart described the coldest summer in memory where every day felt like November. Ultimately this resulted in mass unemployment and famine in the pre-Industrial Revolution world. Famous strategist Von Clausewitz captures it well in his diary while touring Germany:

"Ruined figures, scarcely resembling men prowling around the fields searching for food among the un-harvested and already half rotten potatoes that never grew to maturity"

For those in Germany and elsewhere who were subjected to such devastation, the announcement of assisted emigration to Brazil must have sounded like a life changing opportunity that eventually seeped into immigrant culture venerating the country and Emperor in their lyrics.

I hope this is a useful introduction to German immigration which deos infact pick up at even greater speed in later decades. In particular you see over a million Germans leaving just from 1880-90 in a response to industrial work eventually reaching employment capacity.

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Sources:

RJ Evans, The Pursuit of Power, Europe 1815-1914 (London 2017)

E WIlliamson, The Pengiun History of Latin America (London 2009)

Dukes University, 'German Political Designs With Reference to Brazil', The Hispanic American Historical Review (Durham 1919)