I personally don't believe in scales and other "purely technical exercises". [This is from a classical pianist point of view; other instruments and styles may be different.] There are lots of good exercises in the pieces themselves, namely the difficult places. They are both well-motivated and well-targeted, i.e. you know why you practice them and where you are going to need them. The most difficult thing is probably going to be that you need to stop and really practice them instead of just playing through the whole piece every time.
You'll find some of the difficult places when you practice. To find the others, play through the piece and record it. During recording you'll find new difficult places (because you'll be a little bit more stressed). Then listen to the recording and be surprised: some of the places you thought went quite well don't actually sound that good. :-) So, now you have some exercises.
Another difficulty is that you need to know when you're doing things properly. For this you probably need either a teacher or some good books, videos and lots of reflection. One important thing which can be difficult to learn just on your own is to identify and release unnecessary tension, while still having the necessary tension to be able to play. Now, playing "properly" might not be a priority for you but it will be much more enjoyable (and relaxing!) that way. Note that those purely technical exercises won't help here because you can do those wrong, too!
You can also learn theory this way. You have to identify what you don't know (this can be quite difficult, too!) and then go figure it out. If you do this by getting answers to the whys in addition to the hows, and going as deep as you can, you should be able to get a good grasp of the theory you need.
So does this actually work? I did exactly this for the first 3-4 years. I did have a teacher and that certainly helped in the beginning. Later my teachers rarely taught any technique; I got that from books, experimenting etc. At some point I also started to study theory "formally" but it came quite easily since I already had lots of working knowledge.
The only instrument exercises I have ever done were the scales which I had to learn to pass an exam. I haven't touched or needed them since (in the form scales are learned; of course scales appear in music but there's usually something different, like different fingering, skipping some notes, ...). I never studied the standard piano exercises like Hanon or Czerny. Using this method I got through university and nobody ever complained that my technique is faulty, quite the opposite in fact. I like to think that this is because I never wasted time and energy on useless exercises ;-) Had my teachers forced me to do exercises I would probably have quit quite early.
So, my final answer is this: I do believe it's possible, and even a good way if you have good self-discipline, but having a teacher for some time will be very useful.