Playing the song each time you’re home is being passive aggressive. It’s not an adult solution to the problem.
Here’s what I suggest. Make plans to take dad out to dinner for his birthday. You and him, in a restaurant, for dinner, your treat. Let him know that you really want the quality time alone. Not by playing a song, but by saying these words, ver batim: “Dad, it bothers me that we spend so little time alone together. For your birthday, I would like to take you out to dinner on Tuesday night. Would you prefer _______ or ______ restaurant? I’ll call and make reservations-- is 7 pm OK??” And say those words a week or two ahead of time, so dad has time to clear his calendar.
That way, you’re making a reasonable request in words that convey what you want.
On the subject of being fatherless-- it sounds as though you live in a nice home, and like someone’s paying for college. You’re not an orphan. One of the 12 year olds in my school is an orphan-- her parents and only brother were all killed in a car accident a year ago. Her grandmother moved her cross country, and she started in my school in the middle of the year, knowing exactly one person in the building: her cousin who is 5 years older than she is. In a split second, every single familiar part of her life was lost.
That’s not your situation. You’re not an “orphan.” You mention your mom. You also mention your dad’s wife. They all sound as though they care about you.
And on this subject: " I get to see them a few times a year now tops with college in another state"-- how on earth is that their fault?? I’m guessing you chose your college, not that they sent you off without your consent. YOU chose to move away from home, and now you’re complaining that you never get to see your family?? You’re not being fair here. You would see them more if you lived home. You chose not to live home. And now you’re complaining that you don’t have enough time alone with your family.
Your family is living life while you’re not home. They can’t, and can’t be expected to, drop everything that’s important for the times when you get a 3 day window to be home. Their lives can’t be rearranged on your very limited schedule.
Also, on the subject of having nowhere to go as a family when dad has someone over-- how’s the kitchen or dining room table? Or can a few of you hang out in someone’s bedroom?
You won’t see your family again until Thanksgiving?? Why on earth not? That’s 9 months away.
I think you have a reasonable issue, but that you’re also laying on the drama a bit more than necessary. And you’re acting like the victim; like this is all something that has happened though someone else’s choices. Your dad sounds like a nice guy who may have misplaced his priorities. You’re a college student-- old enough to make a request in simple words.