the truth is that many of the players the club bought were even bought from smaller clubs.
there are exceptions like ballack, sheva or recently lukaku from inter, but most of them were from simpler clubs
as an example,
cahill - bolton
azpi - marseille
hazard - lille
willian - shakthar
matic, david luiz - benfica
alonso - fiori
kante - leicester
remy - lille
oscar - sao paulo
havertz - bayern
werner - leipzig
rudiger - rome
etc etc
only players like filipe luis, fabregas or belletti or deco or pedro were bought from big clubs, but they were the exception.