You asked three questions:
- How long does it take to get to "perfect,"
- How do I, and
- Why should I?
Let's take them out of order, and first take some time to get a working definition of "perfect."
It's true, "perfect" isn't really a useful goal. It's as if astrophysicists ask "how long until we've learned everything we possibly can about the universe?" There's always something more. The saying "practice makes perfect" is absurd, as if you wake up one day and discover that you got your "Perfect Badge" and never need to practice again. Instead, we set ourselves a goal of playing at a certain level, and when we meet that we set a higher goal. E.g., Goal 1 is play all the notes correctly; Goal 2 might be to increase the interpretive and emotional nuance.
But there's nothing wrong with using the word "perfect" if this is in fact what we mean, "perfect enough for my current goal." Perhaps a useful definition for this discussion would be "good enough that I could play the piece for others and both they and I would enjoy it."
Ok, now that we've got a definition, let's start with the third question.
As a hobby pianist, is it worth it? What is gained from this process?
Presumably, you're playing "for fun," for your own enjoyment, not for money, for the congratulations of friends, or to accompany the school musical. Well, the only thing to be gained is new kinds and levels of enjoyment. (As to whether they're worth it, that's up to you.) If we take the analogy of visual art, there could be a certain enjoyment in making a quick charcoal sketch every day, and there's another enjoyment in spending weeks making a detailed oil painting with meticulous detail. Right now you're enjoying the "discovery" phase of learning a piece, and there's a lot to enjoy there. What you're missing is the satisfaction of "seeing" the finished product, as well as many secondary "discovery" phases in which you uncover new facets of a piece you thought you already new. There's also a heady thrill when you attempt something challenging and succeed. The first day you play completely through at performance speed is a bit like the moment from any superhero origin story in which a mild-mannered civilian discovers they have new powers. Young Clark Kent runs through the cornfield and catches air. Keanu Reeves opens his eyes and says "I know kung fu." And there's a related thrill, some time later, from being able to effortlessly kick the %$#@ of a piece that used to be impossible ("Puny god").
What does this process look like?
Unfortunately, between "I know kung fu" and "We have a Hulk" comes a period of drudgery. This is the "practice" that makes "perfect" (if we allow generous scare-quotes). This is the portion that would be rendered as a training montage, but while we're rocking out to "Gonna Fly Now," Rocky still has to chug the raw eggs and run up the steps of the Philly art museum, and it takes him days and weeks rather than minutes. It's not all torture, though; this period is full of mini-milestones with mini-rewards.
For musicians, there's a familiar set of tools and techniques; see these Q/As and no doubt many other posts here about "how to practice." The metronome is your best friend. Start slow, build gradually. Isolate difficult bits, then recontextualize them. Turn off the metronome and work on expression. Record yourself and play it back.
Most of all, even if the only goal is personal fulfillment, you might want to create some informal performances. Performance for others is its own unique joy, and it provides a goal that motivates your work. It's often helpful to book some very small, low-stakes performances before a big one, as part of the learning process (e.g., before a recital, play in a coffeeshop, in a friend's living room, on the sidewalk for change).
And of course, if you're interested, the very best way to accelerate learning is to get lessons.
And finally, the first question:
What is the expected amount of work or amount of time required to take pieces from "I can play this" to "I can play this perfectly"?
Substituting "satisfactorily in a performance" for "perfectly"... well, as I'm sure you can guess, the answer is "It depends." Depends on how hard the piece is, whether you already have a head-start (is it one you played as a kid?), how long and often you practice, and on where you draw the line for "satisfactorily." If you've played for 20 years, you could probably play something out of a beginner piano book "perfectly" with hours if not minutes of practice. A Rachmaninoff concerto might be impossible to attain in a lifetime. But in general? Let's assume that you're at a level at which you play "real pieces," Bach inventions, Chopin etudes, etc. Let's assume that the piece in question is of only "medium" difficulty, and is around 5 minutes long. Let's assume you practice for about half an hour per day, 5 days a week. With those conditions, your experience might look something like:
- First week: getting familiar with the notes. Can play most of the piece through, but have to slow down to muddle through some hard bits.
- Next month or so (more if the piece is hard): Getting mastery over the hard bits. Can play through at a consistent tempo, sometimes without mistakes (if not always).
- Next month: Success becomes more consistent. Start memorizing the piece. If you haven't already, perhaps you start attaining your goal tempo. In this period you pay a lot of attention to expression, nuance, and interpretation (though hopefully you've been working on them to some degree all along).
- Last few weeks before "big performance": In this stage you are simply practicing performing; this is where you put the informal performances. Get used to stopping for nothing. Get used to the act of performing the piece. Get familiar with the patterns of what you think about in which moment, how you need to prepare for what comes next. Find out whether the dramatic and emotional gestures you want are actually coming through to your listeners.
- "Big performance"
- Weeks, months, and years later: Keep maintaining the piece by reviewing now and then. Play it for the sheer fun of being able to bust out something that you know cold. Keep it in your working memory so you have something to show off if called on. Keep discovering new secrets about it and new levels of mastery as the years go on.