Speaking as an Asian, I first got into football via my peers, both Asian and non-Asian. So the first teams I was aware of were Arsenal, Man Utd etc. Loved Henry when I was a really young kid. But I think the FA Cup run helped me get more into Cardiff and also increasing coverage of the EFL on TV and online (bearing in mind I spent most of my childhood away from South Wales, that did help a lot).
I think with the race thing, some Asians have the view that football fans are racist (some of my Asian mates even warn me to be careful at games!) so it must have tricked down. I think also in the past, only the big clubs would have been on TV so the support for these clubs may have trickled down, as some people get to know about football via their peers.
When I was younger, I used to go to City games with my parents, but as I got older I started going with friends. Though with your granddad/dad point, my dad isn't into football as much, but I think via my passion for football both my parents have an appreciation for the game.
It is interesting though how most Asians I know are into the big clubs but I think it might be mainly because of TV/media coverage and getting into football via peers, rather than via family