That is not true for me. I can connect to CCA from both my Mac and PC desktop clients directly with Spotify Connect. Here are screenshots. Spotify is not running on any of my mobile devices.
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I stand corrected. 100% sure this didn't work this way before, but things improved since I used Spotify desktop app last time which was a year ago. Seems they added Chromecast discovery since then, I just checked and my desktop app can see my CCAs.