I feel that I was lucky early in my career. I got a job at an early-stage startup which bootstrapped with agency-style work. I got a generous share in the company, so pumping out high quality work products very much directly contributed to my own interests. I worked a lot, but at the end I was rewarded very well for this when we had an exit.
Since then I have failed to find an incentive structure which actually rewards programming. I've worked inside of standard agile teams, and typically I can handle the tasks which management can prepare for work within a matter of a couple days if not a number of hours. I'm not incentivized to excel in these environments, since I'm not rewarded more for producing output faster than the rest of the team, and there's social consequences for making other programmers "look bad" and increasing management expectations.
In principal I would be happy to just finish my tasks, and then spend the extra time working on projects which interest me, but due to IP clauses in contracts, it's not possible to do this during the many unused work hours without causing issues.
I have tried freelancing, but it seems that most clients want to have a time commitment. So basically they want to have someone acting as an employee, but only for several months vs. a long term commitment. So the same problems arise as you have with the FTE situation.
Currently I transitioned into management, just because I was so bored having to find ways to stretch several hours of work across multiple weeks. I'm a fine enough manager, but it seems kind of silly since I have much more unique value I can add as a programmer of my experience relative to what I can do as a manager.
So basically I am a very prolific programmer, and I am capable of producing a very large amount of value per time on the keyboard. It seems like there must be some way of monetizing this skillset, but I have yet to find it.
Programming is a skill sure, but it does not create value, it creates programs.
Programs may create value, so, the valuable skill is not really being able to create a program, but being able to find some problem, whose solution creates value.
Programming may be required if the problem can be solved by a program, or if a program can solve a problem so much better or faster or cheaper that there is additional value in employing the program..
I'm a programmer, I love programming, but I'm entirely aware that actual programming is not what makes the money. If I had _the_idea_ pop into my head, I'd probably start developing prototypes, that's where my skill would be best applied at that stage.. If they showed promise, I'd likely start full-time development on it, but I'd be looking carefully for the point where I'd be able take on additional programmers and focus more on _the_idea_ and what's around that, I'd definitely hire marketing and leadership too, since those are also not my core competence.. I realize that means that my obvious core competence (programming) is unused, and that's true, my main role would be to try not to screw it up.. Nobody got rich on technical excellence, it's half the time not even a requirement for success.