On my PS4, I had something similar, 100% with all gold missions etc.. Never 'Busted' or 'Wasted' or a single $ spent on taxis.. But that's just me..
Franklin had only ever bought LS Customs and wanting to get the rare 'SNAKEYES' Gauntlet, the 'NIKOB' & 'ROMANB' Emporers, the 'LUISLOPZ' Hexer and the 'PUSSYWAG' Bravado Bison, he bought the Downtown Cab Co. (DCC) to complete his desirable vehicle collection and waited for Raul's call for the missions.. Franklin got the immediate call after buying DCC, welcoming to the business e.t.c., e.t.c. but after that, nothing.. At the same time, Trevor's and Michaels' Property Mission txts, the Random ATM Robberies and Security Vans all stopped appearing for all three characters.. I'd leave my PS4 on for over 11 hours whilst at work, which equates to approx 2 in-game weeks, and I'd find nothing like txts for Franklin upon return.. I'd reload the game and try different things but no txts, ATM's or Sec. Vans.. After finally getting a bit annoyed with the lack of missions from the DCC, Franklin went and bought the Towing Impound.. After receiving the 'welcome' call/txt (I forget which it was exactly), Franklin jumped into the Tow Truck in the Impound Lot and as soon as he began to drive it, the DCC called with a Private Fare mission.. I couldn't believe it!
I reloaded the game-save and rinsed and repeated - same result; Purchasing the Towing Impound AFTER the DCC seemed to unlock/generate the Private Fare calls..
I again reloaded the game-save but this time, purchased the DCC AFTER the Towing Impound, Raul's 'welcome' call came through but no Private Fare calls were then received..
So on my next playthrough, once all the main missions and side-stuff is complete, I'll definitely have Franklin purchase the DCC first..