Which Analog to Digital converter

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.

Curious - what do you mean by “streaming mode”? 
The DA-3000 can be used as a D to A converter to play back your digital files (or from digital inputs)  But it is not as convenient as a streamer which is intended to play files and streaming services. 

I recommend downloading the owners manual to see the details while considering a purchase. 
I did try the portable Tascam DR-100MKIII, and portable Sony PCMD100 before settling on the DA-3000. 24/192 capability was a must have for me. I liked the files produced by the DA-3000 better than those portable devices, and also prefer the convenience of having the device always hooked up in my stack ready to go when I decide to make a recording. 

I am assuming "streaming mode" means the ability for the Tascam to record music directly into a computer and save it as a file as it's being recorded (without having to record onto a USB drive or memory card first and then copying that file into the computer).

But we should let e91811 confirm this for us.