Well today I finished the quest without even owning the bike.
When I started, I thought the mounts count as bike, so I could buy me a mount and then don't worry about the bike and I was right, as I was able to go to the bike road without problems and even do the Subway/Fly Quest there.
However several people have told me, this is not how it is intended to be, so I guessed I should report it then.
As of now I still do not own a bike, I did not pay the 75k and yet I was able to do all the above (probably because of my mount?)
I guess, I will buy a bike sometime later, before I go to Johto, but for now I will leave it like that, so that you mods/admins can take a look into my acc and see it for yourself.
If it is indeed not a bug and works as intended, then sorry for wasting your time.
GorTesK