I have the SAME exact problem. Apple screwed the pooch on this one! First of all, why mess with the way it worked in iTunes? Playlists with my music. Apparently, that's a concept. One would think Apple's "Music" app would contain ALL of the things "music" related, such as play lists. But nope. ONLY the things that are using their paid Music service will show up there, from what I can gather from the internet. The kicker is that IF you wanted to use their service the way they want you to, you will have to upload your ENTIRE music library to Apple's servers. and pay for that privilege after 90 days. If I did that, I'd also need to pay extra on my internet bill because it would blow my usage plan like the US's 1945 Trinity test.
My opinion is that Apple had to solve two problems: {Apple users want access like Windows/Linux users have to files on iDevices} .and. [Apple wants to replace cable TV, making another huge profit]. So, Finder has become what Windows/Linux users have been seeing forever: a half-hearted attempt to let users access their data, while keeping Apple in control of how you can access your data. Opening a Finder window is absolutely useless in this instance and was a standard response from a techie. Had the responding poster listed the steps on how to show you your playlists, then THAT would have been a proper answer to the OP's question. It's kinda like asking someone directions on a trip, them pulling out the state map and saying, "It's on here, somewheres..." as they point at the map. And then walk away.
The Music app is Apple's way of roping you into their media ecosystem, once again. iTunes must not be making enough money for Apple right now, because they sunk a bunch of money and time into this Apple TV fiasco. I have always noticed that when Apple comes out with something new, it's kinda like a skeleton with ONLY loose skin hanging off of it. Over time, it will eventually get "fleshed out" and become a proper entity. The interesting thing is that while you can still see the skeleton, you can also see where Apple is REALLY going with this new product/service. Eventually, the fleshy bits cover this up and hide the underlying skeleton, where upon most users/new users just accept that everything works great and Apple is a super-friendly company.
So, what was once a "semi-easy" thing to do (Apple can't make thing simple) is now a bastardized mess. ALL of those folks at Apple who build and test all of Apple's proprietary stuff and they can't even make it work before releasing it for public consumption. I am seriously thinking about dropping all 5 Mac computers, 3 iPads, and 2 iPhones and going back to Android/Linux. At least I KNOW I'm always going to have to be tinkering, but at a fraction of the price and headaches. AND I can even use my old Macs to run Linux!
Boy, do I miss Steve Jobs right now!!!