Yinsen curiously examined Zangetsu, only then realizing that although Zangetsu's fundamental consciousness originated from Yhwach, he had ultimately formed within Ichigo Kurosaki. As a result, he had inevitably been influenced to some extent. In other words, Zangetsu still acknowledged Ichigo Kurosaki as his "master."
"No wonder he later handed over 'all his power' to Ichigo Kurosaki without hesitation, allowing him to use Mugetsu!"
That made sense. Without that connection, Zangetsu could have simply refused any request from Ichigo Kurosaki.
In other words, Zangetsu had his own preferences, but once his master gave a clear command, at most he could set Ichigo Kurosaki, as his master, some trials according to "the rules of the Soul Reapers." He could not flatly refuse a "reasonable request."
Even so, Yinsen still found it troublesome. He remained inclined to have Ichigo Kurosaki erase Zangetsu's self-awareness and completely turn that power into his own.
Unfortunately, Ichigo Kurosaki and Yinsen were not on the same wavelength. He could not fulfill Yinsen's request and make Zangetsu disappear.
"What? Can't bear to part with him?"
"..."
Ichigo Kurosaki said nothing. Judging from his expression, it probably was not as simple as being reluctant to part with him. Perhaps his thoughts were more complicated—for example... he felt that Zangetsu was a living being, and that he would be making "someone else" die?
As for why he had later been willing to accept Zangetsu and White Hollow's power to use Mugetsu, the situation had been extremely dire at the time, and he himself had needed to pay a considerable price. That probably kept him from feeling too bad about it.
Having figured this out, Yinsen knew that Ichigo Kurosaki definitely would not train in the direction he had planned.
Yinsen was not dissatisfied. Everyone had their own ideas, and he had