I do have the benefit of a PC connected to my living room TV, but even if I didn't, most TVs these days can natively play media from a network share.
There’s wide compatibility with all sort of devices and you dont need to firewall tunnel vpn or do any setup. It’s totally grandma friendly.
Your approach works great for a single user with a tv connected PC. Lets say with your current system you want your parents, right now, to be able to view your movies files. How easy is that to do, and how much technical knowledge or assistance is required?
What is a little nicer about it is that we can hear about something, have it downloaded to the folder that gets indexed, and have it available to play near instantaneously. My NAS also does transcoding if necessary, so that eliminates a lot of hassles around codecs and such as well.
A lot of people take this a step further and avoid all paid services and just use tools like radarr and sonarr to get whatever content they are interested automatically off the high seas and play it when they want to.
The network share is the hard part- well really having the always on server that hosts it- plex/jellyfin/emby etc are just a little bit of sugar on top that make it a nicer experience. And IME, you install once and you are done, there is no maintenance to deal with afterwards so there is little downside.
Ultimately if you just want to browse a filesystem, network shares are fine, but if you want a nice looking front end for that with logos/artwork, descriptions or reviews from the internet, or features that require the files and metadata to be indexed in a DB of some kind, then these UIs come in handy.
Plus they look nice are generally easier for other less tech. savvy members of the househould to use.
Now I just get a pop-up on my phone that Plex has the latest episode. I sit on my couch, hit play on my nvidia shield, and watch on my giant OLED. It's great - and I've been doing this for years now.
Once you go through the initial set up, the UX is fantastic. Far better than anything Netflix, or any commercial provider has ever built.
And for music - Plexamp is an ode to Winamp and is worth it alone. It completely brought me back to the pre-Spotify world of music enjoyment.
And my content is always available from my NAS no matter where I am or what device I have with me.
I do both. But when watching with the family it's much easier to have them a media streaming server than to have to hook the PC to the TV (or projector) and use VLC.
Now at times for whatever reason some media file won't play over the network: dunno why... Maybe it's a blue moon, maybe there's a space in the filename, maybe I didn't respect the directory naming scheme or the file permission or whatever freaking bullshit. When that happens, I put the file on a USB stick, then on to a laptop, then HDMI cable. And that always work. (just had to make sure the TV wasn't using interpolation to 60 Hz or whatever otherwise every movie looked like a soap opera).