Some More thoughts:-
Ichigo is still a Fullbringer. When he was training in fullbring, he first manifested a black reverse swastika that resembled the hilt of Tensa Zangetsu. (see image)
Later the Reverse Swastika fullbring morphed into a sword-like object that resembled shikai Zangetsu (see image)
Then Finally
Ichigo's Shinigami Reiatsu was restored and the result was
This:
And Then This:-
IMO
The diference between Ichigo with Fullbring-Sword, and Ichigo with Zangetsu, is The Increase in Reiatsu, plus the Shinigami quality of the Reiatsu.
In other words I think that
If Ichigo has his normal Shinigami Reiatsu and if that Reiatsu was at full blast, he would automatically manifest Zangetsu when he tries to Fullbring.
That's my theory at least. Now however there is a slight twist. I think that Zangetsu only broke in the first place because Ichigo's Reiatsu was unstable at the time. I think It was unstable because of the unsettled addition of Quincy Reishi/abilities into the mix, and that the form of Zangetsu with Quincy abilities integrated had not yet reach a stable, cohesive equilibrium point, making the unstable sword temporarily fragile.
IMO there would have been no way that Ichigo could have restore his sword by using an Asauchi.
The Asauchi would probably just shatter or disintegrate if he had tried to use one.
Ichigo needs to go somewhere quiet and meditate on his fullbring, he should use his Shinigami /Hollow Reiatsu and Quincy Reishi to form the fullbring. If Ichigo does this I think the result would be a brand new Zangetsu that contains everything he has in him.
In effect this is the same thing that The other Former Shinigami Substitute does/did with his Zanpakutou.
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