Which Analog to Digital converter

The Tascam DA-3000 is a example of a digital recorder that nighthawk is talking about.  It's roughly $899, which is cheaper than the RME.  However, most people that have compared them really like the RME ADI-2 Pro better for sound quality.  With the RME, you only need the $49 VinylStudio Pro software to do everything (and a USB port on the computer).  With the Tascam digital recorder you still need a memory card that is large enough to record the length of music you need and also a memory card reader for your computer (which also connects via USB).  I suppose it all depends on your budget threshold.  I would go for the RME.

That used RME on ebay is gone, but there are several "non pro" versions available, some under $1k.  The "RME ADI-2" which doesn't have that special small LCD screen on the front (it's a step down, but still good.

Additionally, I really don't think you will have the "computer interfacing" issues that nighthawk is talking about.

Another benefit of the RME is that many have rated this as a quite excellent DAC as well, so you will have a really nice DAC to playback anything on your computer to computer speakers or studio monitors of your choice.

As far as fuzztone's promoting the Muffin device.  It is a DSP based phono preamp which makes it very easy to do adjustments and special eq for turntables.  You can get it with S/PDIF output, but then you still need a pro audio interface that has a S/PDIF input so that you can record that music.

I have the Tascam DA-3000 and really like it. Works pretty much like an old school tape deck. Still need to use a computer to add cover art to the files, if you wish. My 24/192 files from vinyl are fantastic. I enjoy them much more than the files available through Qobuz and Tidal. 

In addition to the DA-3000 have a look at the more portable units from Tascam, Zoom, Sound Devices, etc. The interfacing issues I mentioned are mostly drivers not playing nice with Windows 10. If you use a USB stick to transfer files that problem is eliminated.

The Tascam DA-3000 looks very interesting, however does it have a streaming mode so I don't necessarily need to record something to a file before playback?

Yes, it appears that the Tascam DA-3000 does have what you would call a "streaming mode".  You can enable a mode called "ADDA DIRECT" which basically takes any analog audio coming in on one of the inputs and converts and then output that on the analog and S/PDIF digital outputs (it basically becomes a live AD/DA converter).  However, you will still need a digital audio interface to connect it to a computer for recording (such as Focusrite, MOTU, RME, etc.).  This sounds like it defeats your entire purpose of getting the Tascam in the first place. 

The USB inputs on the front of the Tascam do not connect to a computer.  They are meant for a keyboard and a USB drive to record music onto. 

Sounds like a convenient way to record analog sources, but you still need a USB drive or a Compact Flash memory card to copy those files to computer and rename them.  It does look like this Tascam will do DSD recordings if you so desire.