I went through a few phases of this. I cannot say enough how both do matter. A bad streamer can send noise to DAC.
I can also say tons of deals out there used. Hunt and find and save a ton. I got a Sonore Rendu Signature SE and PSAudio Directstream DAC for almost exactly $5000 off Facebook and eBay. They go for about $10k new. Amazing sound.
I am a true believer in the Sonore line. DACs i have only owned a few so not much help other than a recommendation for PSAudio. You will be much happier with the Sound quality by splitting your funds and upgrading both.