Firstly, Zetsu is made from Hashirama's cells, so saying it either way is pretty much the same. It is probable that Obito has been stitched with White Zetsu's body rather than Hashirama's raw cells since it would be quite a clinical procedure to implant those cells and make them grow in desired shape when half of the body is crushed. It makes more sense to just stitch it up with half of a ready-made body. Also, we can see a supposed bulge on Obito's bandaged part, clearly indicating an arm (link:
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). So, it can't be raw cells of hashirama, but a body made out of it, which is technically white zetsu.
It is not necessary to have hashi's cells and use wood release. Only Yamato (or Tenzo) and Madara have ever used Wood Release with implanted cells, and yamato cud only do it because the cells were implanted when he was still sucking a thumb! No wonder Obito doesn't use mokuton, since he has other abilities that won't burn him out or cause him to end up with a tree for half a body (<cough>Danzo<cough>).