Computer or Nvidia Shield as my server?

The reason not to run a Plex server on a regular Shield is internal storage. Plex cannot use any form of adopted storage on the Shield for its database or other support files so, in order to use a regular Shield, you must make sure that everything that causes extra storage to be used, like index files or thumbnails, are turned off. If you do that then a regular Shield seems to work OK. I just really like having index files for movies, TV shows having indexes are not as important to me, so I recommend having a server with enough storage for the Plex data directory, which cannot be moved on the Shield, for movie index files and, for a large library, indexes can use quite a bit of space.

Yes, most NAS solutions are a bit pricey but they are generally quite reliable. If you run a Plex server on one it needs to be a pretty good one with a quality processor or it will bog down if it has to transcode.

I dislike having any wireless devices in my streaming setup but you can generally get by with wireless clients. Servers should always be wired.

I said it in general terms but I will be more specific. Right now if I were to buy a server/storage solution for Plex from scratch I would start with a good but low powered NAS to be strictly used for media storage and a separate device, something like a Shield Pro, to act as my server. I cannot eith good economy change much now with my setup as I have a LOT of storage currently connected to a stand alone computer.
What I currently have, and it does work very well but it is not “best”, is a good PC (That is able to run a Plex server but is being retired from that duty) with 50 tb of USB hard drives attached, All the drives are pooled using StableBit’s DrivePool so what gets shared to my network is one 50tb volume. All my movies and TV shows and other media live in that volume and I have duplication turned on so I will not loose anything if any drive should fail. (Yes I have additional backup but that is another topic) The PC is only used for file service and I keep a good copy of Plex on it (that I activate two or three times a week to keep it fresh) just in case my Shield should prove to not be viable as I have only been using the Shield for a few weeks.
I have the Shield Pro as my server and generally it does nothing but server Plex. I just do not like multitasking devices unless it is really needed.
My main clients are Rokus and I have pretty much retired my Fire TV and my Raspberry Pi as they just do not enhance my experience.

I do not necessarily advocate my setup for others BUT it does work well for me and similar setups are working well for several other people that I have helped get started.

Everybody has to make decisions about how they will spend their money for setting up Plex once they have decided that Plex is the way they want to go for their media. Also they have to decide how their network needs to be setup for Plex’s nest use but I do have some general advice:
Do get the best device for a Plex server you can afford and try very hard not to give it other tasks that it will need to do while acting as a Plex server.
Do get more storage than you need and plan for your data to grow. (Data grows to fill available space MUCH more quickly that most people realize.)
Do get and use clients that fill your needs and desires. Just because a given client is the “best” on someone else’s scale does not mean it is “best” for you. An example for me is the Shield as a client vs the Roku. The Shield plays nearly everything without transcoding and supports virtually all the “fancy” audio and video formats that many people find so important.While the Roku is much more limited as far as all that support goes. But I do not need any of those fancy things and I cannot even hear or see most of the differences. So my choice should be based on interface and there, for me, Roku wins hands down.
Don’t expect wireless streaming to be as smooth or reliable as wired and don’t hook a server up wirelessly.
Don’t expect Plex to do all the work. Get your media named and formatted in a way so as to make Plex’s job as easy as possible.

I guess I have over answered more than a bit but I find that Plex has made my media consumption a lot better than it would otherwise be and i ahve had a chance to reflect a bit about things I should have done differently.