Fundamental religious beliefs? In what religion / belief system? I know devote Christians, Jews, and Muslims that practice Aikido and no of them have a problem with bowing. I personally have very strong Judaic beliefs, and I understand what it means to bow to O-sensei, to my sensei, and to my fellow Aikidoka.
Bowing is about courtesy (demonstrating that I will care for my partner), respect (for my seniors, they earned it) and gratitude (thanks for passing on your knowledge to me). In what belief system is that concept foreign? None that I would hold very high in esteem.
Not to get too much into religious beliefs, but I suspect I know what you are talking about. If people get caught up in the ‘graven image' passages of the bible, they need to reread it and understand it. Idol-worship (which I have heard as the ‘why we can't bow' argument once before) goes well beyond bowing or ‘appearing' to worship something. If your entire life revolves around something (like booze, or TV, or video games, or whatever), i.e. you are obsessed with those things, you are worshipping a ‘false God' make no mistake (because those ‘things' keep you from living a righteous life). If you bow as a sign of courtesy, respect and gratitude you are in no danger of worshipping them. It's about the minds' intent, and God knows your mind's intent (with or without seeing you bow).
Perhaps it would be different if your dojo intermingled Shinto rituals, but most dojos I know of only do that for special occasions (and even then it is a closed dojo session, not everyone kind of thing).
Enough about religion!
Nope, I'd chalk that up to today's attitude towards showing respect to anyone or anything. I mean it's downright hard for people to say ‘excuse me', or even ‘thank-you' to anyone. Common courtesy, respect and gratitude are pretty low now a day. And I wouldn't pigeon just the Americans John, I'm pretty sure we have our fair share of Canadian's that act the same.
Respect and gratitude used to be demanded in the dojo (or you just didn't train there), and because some have now made concessions (adjusting as you said, to the mob rule) it will only get worse. If someone can't bow to his/her partner earnestly, then they do not have the right attitude for any Budo.
It's also quite possible that the person was not really interested to begin with, and just used ‘whatever excuse' to say ‘No, I don't wanna'. In which case, no issue!