I built a home media server because in my future I plan to have my own place and probably won't be able to afford internet. Being offline severely limits your options when cord cutting. So my server currently holds a capacity of 84.2TB, of which 12.6TB is free. I keep TV Shows, Movies, Music, and Photos on it. Because it's a server, I also use it to store backups of my computer, and it's also a storage solution to hold things like eBooks and roms, etc. I access the media on it from a Raspberry Pi that I installed LibreElec on (an OS designed to just hold Kodi). With Kodi, I have access to all the server content. Also on the server I have a series of TV tuners that I have connected to an OTA antenna which picks up all the major broadcast networks (NBC, ABC, CBS, FOX, PBS, CW/WB) as well as a few locally broadcast channels that aren't major networks. I have a total of 5 tuners, giving me the ability to record 5 things at once, or watch 1 and record 4 at the same time. My only problem with this setup is that some of the channels don't come in very well, and signal dropping is a constant problem. Easily remedied with a better antenna or maybe a relocation of it. As for the Kodi side of things, the latest Leia+ version finally brought with it the security requirements to allow premium 3rd party services. I have an Amazon Prime and Netflix subscription, so now I can access those from Kodi. The Pi is still a little under powered to provide a high resolution stream, but 720p works perfectly. 1080p will work with some stuttering, and 4k is not possible (although I think the Pi4 that just came out does support 4k now). If you're not streaming, watching media from the server has no problem playing back 1080p, and Kodi has passthrough so I can actually get the full HD Dolby or DTS sound to my receiver for movies. Kodi is just a glorified media player. But it can do a lot more too. Through "unofficial" 3rd party apps, you can gain access to IPTV services to stream internet tv channels. A lot of cable networks and worldwide channels are available freely. There are also addons which allow you to access internet storage hubs and stream pretty much any movie or tv show you want, by providing a service which is essentially a search engine to find sources online for what you want to watch. My dad doesn't like the stuff I have on my server, so he uses these 3rd party apps on his Kodi in the living room to watch shows and movies. He almost never watches TV on the cable box anymore. If there's a show he wants to watch, he just uses the app to search for it and watches it streamed. He also uses the Amazon and Netflix apps.
A Pi costs roughly $35. With a power supply, SD card, case, remote, and HDMI cable, you can get a fully working system for about $50.
Firesticks, Roku, etc are other ways to get access to online streaming content. You can also use other devices like BluRay players, game consoles, and even your computer if you hook it up to the tv. These all have varying costs. What you need to decide is what content you plan access. I don't own any Amazon devices so I don't know what their cost is and what they access. They might be able to only access Amazon Prime. Netflix is everywhere, but not all devices access Amazon too. Then you have other services you can subscribe to too, like Hulu, HBOGO, or any of the new services like CBS All Access and the new Disney+. These all cost subscriptions on top of the hardware to access them. For me, it just makes more sense to have everything I want to watch stored locally. Streaming is great if you have the bandwidth and the subscription. Or the 3rd party apps that search is a great option, only not EVERYTHING is available, and you have to rely on a good quality host. But there certainly are options out there.
So in closing, it's really up to you. But in my own situation, and through future planning, my setup is this. A home server housing all the media I want to watch, listen to, or view. If there's something I don't have, I'll download it. I watch everything through Kodi, installed on a Raspberry Pi. If it's something I just want to watch, I'll look for it on Amazon or Netflix. If I don't know what I want to watch, I'll use a 3rd party addon to access 24/7 IPTV stations like all-day Action Movies, Sci-Fi Movies, Disney Movies, M*A*S*H, Spongebob, etc. If I want to watch something else, there's the 3rd party addon to search for it online and stream it for me. The system works, and doesn't cost anything more than the hardware and subscriptions to Amazon and Netflix. When I move out and have no internet, I still have my server with all my downloaded content to keep me entertained until I get internet.