Yes you'll typically use an aux send (pre-fade and preferably monitorable on the control room outs) to provide a separate mix to your computer, and then use the main mixer section for yourself. You've actually wandered into the same problem I had when getting my system set up where I wanted a small analog mixer for calls+multiple computers but none of them had this ability unless I went for a large board. Normally I would spec the XR12/Ui12 digital rack mixers for this but their prices have gone up like crazy since COVID. I also didn't want a used analog mixer as those often have bad faders and pots.
I remember looking at the Yamaha you mentioned during my research and the issue with it is that it sends the first two channels direct to the computer as a stereo feed, thus your mic will only appear on one side. Though the same thing happens with most interfaces unless you use a split. Also it sends the signal without processing. Depending on what you're looking for that product may work for you - you really have to trace the block diagrams carefully and mark out the signal paths you're interested in.
If you don't mind an analog aux the Alto ZMX862 also works, though quality is questionable. Both of these options also don't let you monitor the aux send. I think the cheapest option that allows that is the Soundcraft Notepad-8FX which was on my shortlist.
In the end I went with a commercial audio DSP (got mine for around $50 on ebay) and programmed it myself, see my other post here for more info [1].
[1] https://news.ycombinator.com/item?id=34665073