Hi Julie,
Just came across the same issue here. We have been using the tool regularly to extract smaller areas (individual sites) with great success. I just tried a much larger area yesterday and it came out untextured. I reduced my boundary and it worked ok. I haven't had a chance to go back and see if this problem occurs at a particular size.
I think as a workaround you could extract smaller areas, then merge them into a single feature class (if required).
The Clip tool in Pro 2.8 can also directly clip SLPK files out to another SLPK file. Maybe try that first, then use the Extract Multipatch from Mesh tool?
When you say you are using the mesh outside Pro, do you mean in other Esri products or software like SketchUp, Blender etc.?