First of all I want to know why is there not a good description about the algorithm that is used to determine the clipping of a raster based on a polygon. When you look at the slope you can find a very good description about the used methode. This would be very welcome to be use in reports where you have to describe the used methode when analysing data.
A second thing is that it would be helpful if you can choice which part (%) will be used to determine the clipping routine. For example I think the tool now use 50% of the cell to decide if the cell will be extracted or not. When you can choice a percentage it would be possible to get a fully filled polygon with data instead of cells with nodata values.
The second part of your idea is a Duplicate of https://community.esri.com/ideas/3840-clipping-raster-with-vector-polygon.
So this becomes a request to document the logic behind how the clipping works.
You could use the PolygonToRaster (many different options to decide if the polygon is on a cell or not) tool to create a raster filled with 1 and NoData. Then use ExtractByMask or the Times tool to clip.
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