Post independence Zambia was overwhelmingly reliant on Copper exports. Over 90% of exports were copper and over half of government revenue was dependent on copper up until the 1990s.
When Zambia attained independence in 1964 world copper prices were on the rise peaking in 1974. This lead to Kenneth Kaunda's government nationalising the copper industry in 1969, meaning that over 80% of the national economy was in government hands by the 1980s.
The problem with this is that the government became overconfident and spent on lots of infrastructure projects and social programmes. The Tanzam Railway was a particularly wasteful allocation of resources. This was not dissimilar to what other resource-based economies have done during price booms. With Katangan copper added to the mix, Zambia would be even less likely to develop a diversified economy.
Failing to diversify the economy, and with a tumble in copper prices (and commodity prices in general) Zambia would suffer just the same throughout the 1980s and 1990s. Only in 2003 did world commodity prices begin going up, and hence we now have many developing countries finally achieving significant economic growth for the first time since the early 1970s.
Throughout the late 1970s and 1980s Zambia increasingly indebted itself to pay for the bloated state bureaucracy and by 1985 the country was bankrupt and had to turn to IMF restructuring. Adding Katanga to the mix would achieve little to fix this.
On the other hand, Southern Rhodesia's to the south was dependent on tobacco exports, but was a little bit more diversified. However, with UDI in 1965 it forced that country to diversify away from tobacco and to some extent the country was able to do better economically than many of its third world counterparts during the 1980s. However, even in Zimbabwe the tumbling commodity prices throughout the 1990s and early 2000s (1998-2003 was particularly painful for commodity producing nations) probably drove the government there to appropriate white farmers' lands as a diversionary tactic.