My solution to the whole problem is to not even bother.
I consider the issue of holiday greetings, and holidays
in general, to be moot. That's because in my own mind I
have rationalized for myself that they are.
Rather than punctuate my life with moments of cruelty intersparsed with moments of exaggerated kindness, I cruise mostly through the polite middle. Noone cares that I didn't wish them a Merry Fucking Christmas because they already know I give a shit about them. Each day I ask how the people I care about are doing. I am happy for their triumphs, and offer consolation for their problems. That combined with polite behavior, tolerance for others, and a concerted effort to be nice and considerate to everyone I can leaves me content in my human interactions. I see no point in putting additional emphasis on such activities around holiday times. I feel no need to convince people I care. If they still doubt I care after the rest of the year, then oh well; in my mind that means they still have some personal issues to work out. It's not my fault.
Esentially, if you wished everyone a good day each day, all year long, and never tried to make anyone feel sad...then there's no need to do anything special to bolster people's feelings during a holiday. It seems like a straightforward solution to me. Instead of grappling with the problem of how to give that greeting, just remove the need to give a greeting at all. Just wish everyone a happy life all year round.
I do wish I'd stop running into broken people who doubt my sincerity, though. When I say "I'm sorry, that sucks," I mean it literally. Gawd! And the more broken a person is, the more I want to be good to them. But, I digress...
On the other hand, I'm also very devoid of sentiment for traditional things. Graduations aren't important to me.
I don't really care about getting presents on Christmas or my birthday. One of my best friends brings me a shotglass from each exotic business trip he makes, regardless of the time of year he goes. That's more important to me because it proves he actually likes me as a friend, and it's more genuine. Holidays to me seem like everyone just collectively going through the motions. I feel the same way about most traditionally sentimental moments. They're artificial and too disingenuous for my tastes.