However, is my impression anywhere near the reality?
Not with SKA. SKA is a pretty young franchise on the top level. Don't have the history of CSKA at all. They are the Rags of the KHL. Deep pockets, big ambitions, nothing to show for it so far. To put it into perspective: if there is an expensive superstar available on the market(or even if he's not available) the talk goes he'll end up with SKA. 80% of them do end up with SKA.
If you talk about history the problem in the KHL is that many storied franchises never made it to the KHL(Khimik, Krylya Sovetov(Soviet Wings)) or don't have the financial means to be succesfull now(Spartak).
So if you talk about the big names of the past it's: CSKA, Dynamo Moscow, Spartak, Krylya Sovetov. The Moscow teams which lured talent from all over the country. The other prominent team from the soviet era: Khimik Voskresensk, Dinamo Riga, Sokol Kiev, Torpedo Gorkiy(now Nizhniy Novgorod)
After the soviet system was gone the schools that produced the players who would continuously reinforce the Moscow teams became the foundation for the new powerhouses which now thrive in the KHL such as: Omsk, Magnitogorsk, Yaroslavl, Nizhniy Novgorod, Togliatti(fell apart, but supposed to be in the KHL soon), Chelyabinsk(took them longer to rise due to finance troubles, and the talent drain is the killer).
The Moscow teams on the contrary fell off the face of the Earth in a way. Until Dynamo and now CSKA found some financial support(well, Dynamo actually always managed to be good even through the rough times). The thing is they still have the schools that produce talent along with spartak which is actually bankrupt but their MHL team recently won the Kharlamov Cup(MHL top trophy) in game 7 against CSKA youth "Red Army".
Now Dynamo and CSKA have the money to compete and Spartak is in limbo.
And then there are the franchises which of course weren't built from the scratch, but actually never were near the top in soviet times. That would be SKA, Salavat Yulaev Ufa and AkBars. With financial support those teams not only built KHL teams, but also youth hockey schools which are now producing a decent amount of talent.
As you can see, due the upheavals of the 90's the is no consistency with all this. New teams grew to prominence and some faded.