Here in Australia, I suspect that a big part of the aversion to the Happy Holidays greeting is that it has its origins in the USA. There is - and this began long before the presidential election in November - a real anti movement against the encroachment of American cultural mores becoming embedded into Australian culture. For myself, a big part of that is the plethora of American television shows - which I see more of because Dragon Dad loves a lot of them, while I really don't. Odd ones I do, but for the most part, I'd love to see more locally made programs, and definitely more of the quality television we used to have from the BBC which appears to be one of the losses to the incursion of American TV. That's probably the most overt thing, but there are many more, not to mention a sense of unease about some of the political decisions that have been made over time due to the terms of our alliance with the US.
There is also a feeling among many people I talk with that political correctness is swinging to extremes, and there are things that perhaps need to be discussed that aren't even broached anymore for fear that people will be called out for saying the 'wrong' thing. So, there are two things going on, really, and as far as popular culture is concerned, it does seem to come out at the end of the year when the Christmas frenzy starts to go overboard.
For myself, I'm Jewish, so I don't celebrate Christmas. So, for me, being wished Happy Holidays definitely feels inclusive. It means I don't find myself in a position of feeling as if I ought to educate people about Chanukah - which this year, begins the evening of the 24th of December. I don't feel as if I have to insist that, no, it's NOT the 'Jewish Christmas' on the odd occasions when I make the choice to explain and hit someone who doesn't know much about Judaism. I do feel that the person wishing me Happy Holidays has thought about the fact that Christmas isn't the only religious festival that happens around this time of the year.
People do ask what my family does 'for' Christmas. Well, we don't do anything - see above comment about not celebrating Christmas. But we do get together as a family, depending on who's around and what various family members are up to. This year, Dragon Dad's father is visiting and we'll be having lunch at a restaurant that we like. We've discussed perhaps inviting those friends that are free to come and celebrate lighting the first candle on the chanukiah and eat latkes with us - we haven't got as far as doing anything about that yet.
So, whatever your particular tradition happens to be, I'll wish you Happy Holidays, and hope that by using a greeting that covers a variety of occasions that it will be received with the goodwill with which it is offered.