Mapping those connections are the least you have to do (and I think that should solve the problem in that case, thank you).

Mapping sidewalks separatly is a long time discussion in the community but the most say that only the existence of a curb is definitevly not enough to map a separate way. In my personal opinion I would also avoid separate ways even if there is a parking lane or trees because if there is no car parked you a are able to cross the street (even between cars) and for me tree areas are only relevant if they have a certain size (so you just could not walk around them).

And there is another (maybe minor) routing problem: Because the separatly mapped ways do not have names, a router could not tell you "Turn right into Steppenbergallee", instead it would only say "Turn right". Adding the street name to those ways could be a solution but is highly disputed.