Yes most bikes share the engine and gearbox oil , but I have a automatic truck and it has separate engine and gearbox oil , I have just changed all the oils on my truck so I have truck head on and not bike head , sorry if I confused you .
I don't no why a dirty pick up filter would affect the gear selection , but it DID help me when I cleaned it .
Sticking clutch plates due to bad oil (contamination) due to lack of servicing by previous owners was my problem .
I had clean ENGINE oil but my clutch plates still stuck together on first start up/gear selection . Gear selection was fine afterwards .
I dumped the clean ENGINE oil , cleaned the pick up filter and put fresh ENGINE oil in as well as another oil filter .
Selecting first gear , first start up improved ,
Sorry , only saying what was wrong with my bike .
I will now go and sit in the corner and be quiet .
Good luck with getting it sorted .