Help me find my ideal Stereo Integrated Amp

You might want to take a look at Advance Paris X-i1100 integrated amplifier. I think it checks most of the boxes that you're looking for. Good luck on your search mate.

+1 on Anthem STR integrated.

My son has one and it’s great sound with very good DAC and phono pre. He streams from a node but uses the digital out b/c he prefers the Anthem’s DAC. Has a boatload of digital inputs and RCA XLR analog. Around half your budget but Stereophile Class A. Easy to set up and use. Worth a listen. 

True the Anthem has no streamer, but with tech formats changing so fast it's probably better to use a separate. Your budget has plenty of room for a separate streamer anyway. 

The Hegel H600 is a lovely box that should do it all and well. 

Maybe underpowered at 150W into 8 ohms, <1% THD; 280W into 4 ohms, but I’m going to suggest Aesthetix Mimas.

Most integrated DACs are okay to subpar, so it usually best to get a separate DAC. Integrated phono is usually minimally satisfactory. But the Aesthitix Mimas optional DAC and phono cards are sonically superior. Aesthetix has been around a long time with a great sonic reputation. Missing is the streamer option, but extremely few audiophile quality integrates have it.

The Hegel H390/H590 has good DACs, the H600 an notably better DAC, but don’t offer phono or streaming.

Other integrated choices at the 10k price point the DAC sonics would be notably inferior, maybe a good trade off if the OP decides that features (having DAC, phono, streamer in one case) are more important than best sonics.