Hokay! Allow me to say what I'm thinking here!
First the known:
Ichigo's father was obviously a shinigami of of one sort or another
IIRC Ichigo's reiryoku caused Orihime and Chad to gain powers somewhat akin to the people with Fullbring
Ichigo has hollow powers because he had actually partially transformed into a hollow after he lost his powers the first time
Ichigo has not used his hollow powers since before the fullbring arc
Now the unknown:
True source of Ichigo's powers
Now my theory!
In the beginning when Rukia awakened his spiritual powers they manifested in shinigami form because she used her Zampakto to awaken them, his affinity for being a shinigami being helped along by his father. Had they awakened on their own, they would have most likely manifested either as a fullbring, or quincy powers. Because of Ichigo's enormous reiryoku he was able to reawaken his shinigami powers to overcome the hollow transformation, only because he was reaching for them, IIRC there where other strings he could have followed aside from the shinigami one. So upto the fullbring arc he had always been reaching for only his shinigami powers. Now lets just put fullbring aside for now as it was only a plot device to give Ichigo his powers back without doing the same thing with separating the sould from the body again, like Uruhara did. The only way for Ichigo to get out of that jail he was in between worlds would have been to do 2 things, awaken a new power, or use his hollow form, but as he has not used his hollow powers in quite a while I think Kubo forgot about them. So that leaves us with one option left, new power. But Kubo doesn't want a completely new power, most of Ichigo's powers have already been seen in one form or another. So for the new power, you have Kido, or you have Quincy powers. As Ichigo is not the type to recite long lines of poetry for an attack, nor has he been trained in Kido, that just leaves Quincy powers, which would make more sense for breaking out of a Quincy spell. Now since he says to take Ichigo back and they can resurect him as one of them, it leads me to believe that there is some higher connection between Quincy and Shinigami, most likely that Quincy are actually an evolved form/different type of shinigami that came about when a few of the VERY old shinigami decided something a little more drastic had to be done about hollows.
Well that's it for my ramblings, hope you don't hate me so much!