Different culture and lack of banking, checks were not a thing here and it was a cash economy. I'll share a funny story; when I started working in Russia, I took 80% of my salary in dollars (not allowed by law) and 20% in roubles for everyday expenses. I used to go to one room for roubles that had windows on three sides so everyone could see. Then go to a different room with no windows to collect my dollars. Nowadays, most people I know have direct deposit, but salaries in cash are still common. Not sure about players, I would guess for rich teams they would direct deposit and poor teams via brown paper bags.Many of the former NHLers that have played there have said that they were literally paid in bags of cash. Not by the KGB directly, but people also say the KGB and Kremlin are closely tied to the KHL.
So, you are pretty much right.
As for KGB, or FSB as they are now; any state owned company have a few of those guys working there to keep an eye on things. Most KHL teams are sponsored by state companies, so there will be some overlap. Some teams like Dynamo, CSKA, SKA etc are directly tied to government departments.