Why does Adobe Premiere increase video size even when nothing has been done?
You cannot do this with Premiere.
The media info of the recording tell us that you need a maximum video bitrate of 0.108 Mbps to achieve 15 Mbyte. However Premiere's smallest available bitrate is 0.19 Mbps which would yield a 23 Mbyte final video (including the audio) in theory which is pretty close to what you see. As far as I know you cannot get below that.
If this bothers you, contact Adobe for help. Technically there is no obvious reason to stop at 0.19 Mbps. Tools like ffmpeg can work with even lower bitrates with this codec.