Hello,
A bit late, so I'm not sure if this is relevant for OP, but it might be relevant for anyone else who ran into the same issue and stumbled upon this post.
Context: Recording games on OBS Studio on my Mac as well. Muting or disabling the Mic from the Audio Mixer would just record screen captures with no audio. The Global Audio Device for my Desktop Audio (in the Audio section of the settings) is "[Device not connected or not available]," which I thought was the cause of this issue. I managed to fix the issue without adding a device to record desktop Audio.
Step By Step Guide (with screenshots)
Step 1 - Make sure Output Mode is set to Advanced
Open OBS Studio's preferences. Click on the Output section, and change the Output Mode from Simple to Advanced if it is not already set to that. Click the Apply button in the bottom left corner of the Settings box, and the OK button in the bottom right corner to save these settings.
Step 2 - Separate the tracks the audio is being recorded on
Open the Advanced Audio Properties (click the button with the two cogs in the bottom left corner of the Audio Mixer section). Make sure the Desktop Audio and the Mic inputs are on different tracks by unchecking any boxes for tracks they are both on. Audio input capture is not relevant for this issue. As long as there is at least one track which records from the Desktop and not the mic, this should work.
Step 3 - Select the Desktop audio track for recordings
In the Recording Tab of the Output section of OBS Studio's Settings, make sure to omit any tracks that have the Microphone input selected. In Step 2, I chose Track 2 for Desktop Audio, so in this step I must also select Track 2 for the audio tracks in my recordings. Click the Apply button in the bottom left corner of the Settings box, and the OK button in the bottom right corner to save these settings.
And viola! Your recordings should only have the audio from the desktop and not the microphone. If you want to include audio from your Mic you will have to add the track that it is recording on to the Recording Settings section of Recording Tab of the Output Section of OBS Studio Settings.
Hope this helps!