There is no a single center of origin for some genera. Some of them have a polyphyletic origin. May be they have centre of origin in wide sense? (Palaearctic, Neotropic etc.)

The belief that there is a centre of origin is a spin off of linear thinking. While it is one thing to state that a genus is Oriental, or Nearctic, it is quite another thing to assert that the centre of origin of, say, the Pieris genus, is in the Caucasus. In a few cases, one might have deep insight and might suggest a small geographical area , for example, Rhaphus probably had a centre of origin on Mauritius, but there is no fixed set of factors which, when examined or analysed, will yield the imaginary 'centre of origin' of a genus or other group.

in the classification of organisms, a genus is  consisting of at least one and often more than one species. Do not  confuse the actual distribution, the center of dispersion, the possible multiple centers of diversification, and the result: nowadays the diversity in species. In oldest organisms (i.e. arthropods,) some representatives of  actual genus considered as "gondwanan"  are found in North hemisphere: without fossils you can speculate on "this genus originated from southern continents" . But  in fact these genera may have been in Europe, but they are now extinct, but survivals are  still in southern continents, the secondary place..... What you can find more easily is the most recent center of diversification ! and don't forget that if the concept and the definition of a species are usually simple, the genera and their definition are often a subject of multiple discussions between specialists . So "where are the fossils?"