Did the Haber-Bosch process enable the population explosion?
There are a number of factors that went/are going into the population explosion, food is only one vaccination and antibiotics are a big factor as well.  within food fertilizer is again only one of many factors. Better crop strains, mechanized and better farming techniques, soil science, pesticides, refrigeration, and a dozen other factors contributed.  Keep in mind artificial fertilizers get less important as you go back in time as well so calling it the detonator is incorrect and misleading. 

Now when considering crop rotation, crop rotation has pros and cons, there are several nutrients plants need and rotation only helps with some of them, and it relies on having compatible crops to rotate which not every crop or environment has.

Keep in mind the piece wikipedia is quoting is not a peer reviewed paper but an essay in nature, which explains it is referring to several modern countries that fertilize almost exclusively with artificial fertilizers. It is not claiming they cannot use other fertilizers.