It depends on how the streamer and the DAC are connected. In the stuff below the capitalized DAC refers to the whole component and dac revers to the innermost guts - the chip that turns the digital data into an analog voltage/current.
Critical to sound quality are the accuracy of the digital to analog conversion, the filtering of that process (which initially yields a "staircase" waveform) to create a smother analog signal. These are not affected by the streamer - assuming it sends the correct data to the DAC.
And then there is the timing of the digital to analog conversion steps, when digital data is strobed into the dac chips by a clock signal. If that strobing is not done at precise intervals - phase noise or jitter - the waveform is distorted, so a key contributor to SQ is the accuracy of the clock, not just its frequency but how precisely regular it is. This is why DAC designers (and Streamer designers) go to great lengths to ensure accurate and stable clocks.
If the connection is IIS (I2S) then that clock is transmitted from the streamer to the DAC in the cable. If your streamer's clock is better than the DAC's, and you have a first class cable, your SQ will improve. Similarly with Coax and Toslink the clocking is to some extent dependent on the streamer, hence SQ will depend to a greater or lesser extent on the streamer in addition to the DAC.
If you use Asynchronous USB the SQ is probably unaffected by the streamer as the data is read by the DAC, on request, into a first-in-first-out buffer (a chunk of RAM) and is strobed into the actual dac chips by the DAC's clock. So the analog output is isolated from timing errors (jitter) from the digital source.
The author of the YouTube makes this point at about 7:45.
I have done an experiment on my DAC, using USB, if I "pull the plug" on the USB cable there is a small but just there delay before the sound stops as the data has been transferred into the DAC's buffer.
There is also the issue of (loosely speaking) electrical noise, but the same argument applies - the DAC should be implemented to isolate its analog output from glitches on the digital cable, power cable and ground. If changing your streamer affects the sound IT IS YOUR DAC THAT IS DEFICIENT.
I am probably going to get yelled at for writing this but IMHO put your $$$ in the DAC. Make sure it supports USB. If that component was truly well designed, engineered and implemented, and you are using an OK USB cable putting $$$ into the digital componentry and wiring before that is probably not going to change your listening experience - assuming an OK streamer and OK network.
My implementation is somewhat extreme, the DAC is the Esoteric K-01XDSE, with a Stanford Research FS 725 Rubidium clock - accounting for 90%+ of the cost of the digital equipment, and the sound is superb, the Esoteric engineers have gone to extremes to create this isolation, perhaps others have not.
I have tried changing USB cables (Cardas vs a cable than came with a printer), Streamers (Aurender vs Eversolo T8), Internet connections (wired vs WiFi) - no difference. Using the DAC's internal clock vs. the External clock - noticeable soundstage differences even though the DAC's clock is excellent. Using Coax or Toslink instead of USB - a definite drop off.
Once the signal is in the analog domain it is a different story, everything on or near the signal path matters.