The example you cite where Morihei Ueshiba did not like to be called religious, could it be that he simply saw reality a bit differently than us to begin with? This is the man who saw the kami as tangible things and believed he was channeling them, no? He may regard those beliefs in the same way that he believes water is wet or the sun gives light or that mountains are fookin heavy, that they are inarguable aspects of his reality, not religious tenets of faith. So the question of whether one is Buddhist, Catholic, Protestant, Wiccan, etc. would be moot, because he could still see those inarguable tenets of his reality as manifesting regardless of which faith any practitioner chose, and that they would still be confronted with them. Just because he didn't see himself as religious doesn't mean he wasn't.
If I remember the quote, Ueshiba most definitely got irate when people called him a religious man and replied he was a man of Budo. I can't find it at the moment.
There are these, though:
"When anybody asks is my Aiki budo principles are taken from religion, I say ‘No.' My true budo principles enlighten religions and lead them to completion." (http://doveraikido.com/about_o_sensei
Aikido and the Harmony of Nature by Mitsugi Saotome. Ueshiba quoted as saying, "Aikido is not the way of weakness or escape, for obviously Budo belongs to those of strength and skill. Yet the Way must lead to a world of mutual concern and respect for one another."
Ueshiba is talking about his aikido being a Budo of strength and skill with mutual concern, not as a religion. Ueshiba viewed his aikido as a martial art and not a religion and he viewed himself as following Budo and not being a religious man.