It is odd that "Merry Christmas" became offensive to some just as its becoming entirely watered-down and innocuous was complete. Until about 1980 it was just the standard well-wishing, and few even among the nominal Christians had any conscious thought that it meant, "I want you to have a joyous religious holiday celebrating an important event in spiritual history." It was generic, in that if a person replied, whether in humor or irritation "I'm Jewish," the reflexive reply would be "Oh? Well Happy Hanukah then," with the intent of "Same thing, didn't mean to be insulting." That the specifically Christian history was regarded as generic may be the bulk of why people were offended - their little pushback against a majority culture that had assumed too much about itself.
Odd that "Merry Christmas," rather than becoming even less Christian - like "good-bye," or "God bless you" now carries a touch of aggressiveness about it. Whether one says that or "Happy Holidays*" is now a cultural statement. Come to think of it, both could be any of a few distinct cultural statements:
"This is what management told us to say, it's policy."
"This seems the safest thing to say these days."
"I want to make sure no one feels left out."
"I am emphasising that we should not say Merry Christmas"
and likely a few others
"This is the traditional greeting, and I like tradition."
"There's still a lot of us Christians and we'll say it anyway."
"I really want you to have a spiritually uplifting season."
"I like announcing in little ways that I follow Jesus."
"I haven't really thought about it. It's just what I've always said."
There are often shades of a few meanings in any one person's usage.
Those of ill will assume the worst about others. Those of good will assume a better meaning either way. "Peace," in an early Christian context means "agreement, accord," related to pact more than "absence of war." Ereine in Greek refers to the presence of something, not an absence.
So Peace on Earth to those of good will. (Does that imply "The rest of you can take a hike?" It's hard to tell these days.)
* "Season's Greetings" occurs largely in print, not speech.