JR is big, but 62% of passenger volume is not JR and that remaining 38% is split by 7 companies

Further, in the big metro areas, the private trains do just fine.

JR East is #1, Tokyo Metro is #2, JR West is #3, Tokyu is #4, ... the next JR, JR Central is down at #9 with #5 #6 #7 #8 all private. Tokyo Metro is private, Toei (is the city run subway, it has 4 lines as is far down the list).