I don't really care to be honest. I think that either Happy Holidays or your personal greeting is fine. Personally I have zero religious affiliation. I switch between "merry christmas" and "happy holidays" without thought because they don't have meaning to me on a spiritual level. Christmas has over time become both neutral and associated with Christianity, while Happy Holidays is neutral. I don't see anything problematic with governments or businesses sticking with something neutral; this is often just what comes with territory of serving a broad audience
But still, to speak directly about what's in the OP, I agree. To have someone greet you during "the holiday season" in a way that reflects their spirituality is not necessarily some veiled attempt at establishing cultural hegemony or insulting you/your beliefs/your mom. When someone greets you like that why would you assume the worst or react negatively.
I don't agree, I think it would be silly for anyone to be offended by any entity celebrating their culture/religion/what-have-you. You can change it to something plain and blanketing, but I feel then rather than 'including' everyone, it just excludes everyone and becomes meaningless.
If I lived in another culture, I would expect them to celebrate that culture freely, rather than having to change their wording because it might offend some dickhead.
Honestly, the only people who ever fight for the abolishment of 'Christmas' phrasing are dull white people who race to be the most 'tolerant' so they can smell their own farts and pretend they are superior.
You were on the naughty list this year, weren't you. =P
America hasn't been about cultural uniformity for a long time, I think us having this discussion speaks for itself.