Firstly, saying China had a destitute economy and archaic technology from 1945-1990, with a GDP lower than 99% of the countries of the world, is simply wrong. In 1990, the Chinese GDP per capita was nearly $2000 per capita, which was much higher than, say, India's. Most people in the cities had TVs, businesses had vans, factories were full of machinery, people bought factory-produced clothes rather than hand-made clothes, children had plastic transformers to play with (admittedly they were knock-offs, but as a 4 year old I didn't care!), many houses, especially in the city, had washing machines (though not very good ones). Admittedly few people owned cars, unlike today.

Going back to the 50s, China also had very high growth, almost doubling its GDP per capita between 1950 and 1960. Admittedly, it's GDP fluctuated massively through the chaos of the 1960s, only growing slightly from 1960 to 1970, but it started to recover in the 70s, and then with the "reform and opening up" throughout the 80s, China had very high economic growth. China's GDP in 1990 was over 3 times higher than its GDP in 1980. And approximately 9 times that of its GDP in the early 50s. (These figures are all real-term figures; due to inflation, China's GDP in 1990 was actually 14 times that of its GDP in the early 50s, but that's not in real terms.)

Then, of course, as you note there's other elements to its importance. The massive population of the country (about 1/5th of the world, in 1990), and the geographical location (near Russia, near Japan, near Vietnam, near Korea). I wouldn't say there was many historical elements though: European experiences with China were of what they saw as the archaic, corrupt, and easily exploited, creaking Qing Dynasty, not the cultural and technological superpower of China's past.

There were 3 other key points though.

1. China had, and continues to have, the largest army in the world. Not the best equipped or best trained, but the largest. In any land war against its neighbours, China was likely to come out on top. (China's weakness was its navy, which was deemed incapable of a successful invasion of, say, Taiwan.)

2. Politics. China was the Soviet Union's most powerful ally, and protected the USSR's south-east. It was also the backer of a number of other Asian communist states and rebel movements, like the Khmer Rouge and North Korea. In fact, the reason that North Korea was able to survive the Korean War, is because America and its allies did not want to fight a war with China. In the 1970s, the Sino-Soviet split left America with an opening to weaken its Cold War foe. Nixon went to China partly to weaken the USSR by exacerbating the split. This was during the time of the Vietnam War, whose Communists were supported by China and the USSR. Thus, eroding Chinese support so that only the USSR (which was, importantly, not literally right next to Vietnam... unlike China, which was) supported the Viet Minh was extremely important for America.

3. Nukes. Nukes. Nukes. The People's Republic of China had nuclear bombs, and the capability to use them on American allies such as South Korea, Taiwan and Japan.

Now saying all this... I'm not sure I actually agree with your original premise. In 1990, yes, China was recognised as a major world power, but in 1960, I'm not sure I'd agree with the idea that China was considered a significant world power. Big, somewhat signifcant for its geographical position and big population, yes. But China (by which I mean the People's Republic of China, ie mainland communist China) wasn't even recognised by the UN until 1971.

Nixon went to China in 1971... but why not before? You said why didn't he go to Africa or Brazil.

Well, Roosevelt went to Brazil, and the USSR, and 7 countries in Africa, and Iran (and other places in the Americas and Europe), but not China.

Truman went to Brazil (and other places in the Americas and Europe).

Eisenhower went to Brazil (and other places in the Americas, Africa and the Middle East, as well as India).

But no President had gone to China before Nixon.

Nixon didn't go to China because one of the world's main powers (though still a significant one). He went there because it was an opportunity to weaken the USSR.

I'm not saying China was insignificant. It's nuclear weapons, geo-strategic position near Western-aligned nations (and, in the early post-war years, Western-ruled countries like India and Vietnam), and massive size/population/army, ensured it couldn't be ignored. But generally speaking, in this period it was not a key player on the world stage.

The question at the end, I suppose, is what exactly do you consider a "Great Power?" It wasn't on the UN permanent security council until 1971, for example.