Under FAA rules, simulator training is not a requirement for any certificate/license. Only a certain amount of simulator training can be substituted for training in an actual aircraft for each certificate. Even then, only simulators certified by the FAA can be counted towards training time. The cost of the hardware and complexity of the software necessary to be good enough to receive FAA certification makes only high level FFS devices financially advantageous for airlines. These devices compared to the average home simulators is like the difference between 1970s TV night to holographic day.
Then, the question is if the allowed simulator time makes sense for the General Aviation pilot. Part 61.109 allows 2.5 hours of sim time to be substituted for actual flight time towards a private pilot certificate. That number doubles if the training is done in a Part 142 training center. By the time you get to the 1500 hours required for an ATP, ...
Not more than 100 hours of the total aeronautical experience requirements of paragraph (a) of this section or §61.160 may be obtained in a full flight simulator or flight training device provided the device represents an airplane and the aeronautical experience was accomplished as part of an approved training course in parts 121, 135, 141, or 142 of this chapter.
If the time requirements for aeronautical experience were arbitrary to the need for training, a flight simulator can be a viable option. A realistic AATD (Advanced Aviation Training Device) would be affordable for a General Aviation flight school. Compared to the millions of dollars of a full-motion FFS (Full Flight Simulator), an AATD would be less than $100,000. Most flight schools in my area would charge a student roughly half the hourly rate for an AATD that they would for the typical 4-seat, 6-pack, single-engine, fixed-winged trainer aircraft.
Unfortunately, the lack of realism in an AATD makes it a valuable tool for procedures training only. An AATD lacks the feel and feedback of real flight. Its value is only as good as the student, instructor, or pilot’s diligence to following the procedures. To gain the most training benefit from an AATD carry out the flight using flight planning, flow checks and checklists to their fullest. Role-playing ATC will further add to this benefit. Treat it like a real flight and not like a game.
Anecdotally, I have trained on sims ranging from full-motion Redbird BATDs with custom G1000 bezels to stationary AATDs with fully functional instruments and controls, home sims on personal computers to multimillion dollar full-motion, gimbaled FFSs at the American Airlines Training Center. Except for the AATC FFS, none of them had the realistic force feedback of a real flight. All but the FFS had software sophisticated enough to mimic landings or wind interaction with enough realism. And, the FFSs are far too expensive for general aviation use. Actual flight time in a GA trainer aircraft would be cheaper.