I worked on this exact issue for several years. In fact, TSA did test a system connecting TDC stations to the airline reservation system back in 2002 at LGA, and it worked great. Operationally, you could have checked in with only your FF card.
The challenges in rolling it out -- in addition to sheer bureaucracy and Congressional micro-management -- were cost and complexity in dealing with legacy systems across both the industry and the government. Eventually, Secure Flight did link them up, but TDCs obviously do not have a direct electronic connection to Secure Flight.
Mobile Boarding Passes use IATA's M-BCPB standard, which includes fields for encrypted information. The scanners that TDCs use for MBPs are equipped to work with that encrypted information in a reasonably secure way. Forging an MBP to beat Secure Flight is NOT easy.
As for IDs and paper boarding passes, I'm not going to go into the history of all of that because my head might explode...