Suggestions for Server Set Up?

Hello! This is my first post, so let me apologize if I’m long winded. I need suggestions, though, if anyone is willing to offer any!

Let me preface this by saying that I’m definitely not a networking expert. However, I do consider myself at least slightly above average with computers, and with the proper tools (as well as my own research and time given), I can generally figure out whatever it is I need to do, whether it be with a command prompt (Terminal, in my case) or general hardware/software set up.

I have been using Plex for quite some time now and I love it! I currently have it set up in a way that makes it a long winded process to start the server and make sure all my files are available to stream, though. I will outline my current set up below:
I have my media files saved on my Airport Time Capsule. I have the Time Capsule set up to be used as a network disk of sorts (NAS?) with Plex searching the folders I have created on the disk for my media files. With the current limitations of Time Capsule, I guess, I currently have to open Finder on my MacBook Pro, connect to the Time Capsule, then I have to load Plex and hope the files don’t show up as unavailable. If they do, it usually leads to me having to close the app, verify the disk is connected and available, and sometimes re-point Plex to the folder on the Time Capsule. Of course, I’m also using my laptop as the server, meaning I have to have it out and the screen open and connected to the internet for the Plex app to load anywhere else. And since it’s my only laptop, I take it with me a lot when I leave the house, meaning the Plex Server isn’t functional when my laptop isn’t there, on, logged in, and connected to the internet and Time Capsule. So basically, every time I want to watch a movie on Plex, there’s at least 5-10 minutes of set up involved before I can even use the Plex app.

I’ve been in the market for a better set up, and have been doing a lot of research lately trying to figure out the best option. I had initially settled on the Nvidia Shield, but I’m beginning to think there might be superfluous features (like the Android OS) that I don’t need that will end up costing me more money.

So I guess I’ll tell you what I want, then I’ll list what hardware I currently have and what my budget is hoping to be. I would really love some suggestions on the type of set up that would be most beneficial for me.

What I want in a server:
Once the server is set up, I would like it to be just kind of put in a closet somewhere, connected to my router, and left to do it’s thing. I’m fine with putting in a day or so of set up time if need be, but having to jump through these hoops every time I want to use Plex is becoming a tad cumbersome for me.
I would like to have a decent amount of storage for all my movies and TV shows. At least 500gigs would be preferable, but there’s a little wiggle room when considering the budget. I would also like the server to be accessible remotely. That way my mother (who is definitley NOT tech savvy) can watch her shows in her apartment from my server, and I might be able to watch away from home as well. I would like to be able to use my Apple TV or my SmartTV to view my media, meaning I don’t have any need or want for a server that also acts as an app.

What I currently own:
I have a 4K Samsung SmartTV. I only use the Plex app on the TV when we want to watch something in 4K. Otherwise, I use my Apple TV for pretty much everything. I also have a Chromecast in another room, and my mother has a SmartTV with Roku built in, and a lower midrange Android phone with the Plex app on it.
I, of course, also have my Airport Time Capsule. It has 2 TB of storage and also acts as my router.
Then there’s my MacBook Pro and my iPhone X, both of which will be used to watch Plex videos but will not be kept at home all the time.

I would like to keep my budget around $150-$170, with maybe the option to add more storage in the future if I come into some extra cash?

Final notes/questions:
I do have a wide variety of file formats in my library. My preference for video files has always been MKV, though I’m not completely against converting everything in order to maintain the set up I want.

When it comes to the possibility of having to encode while streaming, do I need a server like the Nvidia Shield? Do my Apple TV or SmartTV have the capability RAM/GPU/programming-wise to convert files if needbe? Would an NAS work as a Plex server without an out-of-the-box graphical interface like the Nvidia Shield has? I’d rather spend my money on more storage/streaming capability than something “easy to set up” or with an OS pre-installed.