For all we know, Ashura did not have the Mokuton
Hashirama must have had a special gene
He did, Indra and Ashura were both influenced by the Juubi's power because their father was its Jinchuuriki, in addition Ashura inherited the body of a JuubiJin so would logically have it's KKG at the very least.
In addition, Senju and Uzumaki both descend from Ashura therefore should have his abilities equally divided amongst them. He can use chakra cloaks like an Uzumaki so should therefore be capable of using Mokuton like a Senju. One clan inherited Gedo traits, the other clan inherited Juubi traits.
Furthermore, think of Kishi's emphasis on "equality". Indra has Fire and Lightning affinity, eventually wielded Enton. Ashura had Earth and Water affinity, which is direct opposite of Indra's affinity, and eventually wielded Mokuton, the only element capable of nullifying Amaterasu.
Also, Ashura didn't have Kurama for regeneration feats but healing is supposed to be his power, only other tool capable of granting regeneration is Mokuton, as proven by JuubiJins.