One thing that might help against racism is doing what some TV, movie, and theatre producers are doing: They are mixing up the races, as if the race of a character is totally irrelevant. It is never even mentioned. So, in Bridgerton (Netflix), people who are any and all races are having mad, passionate love affairs, and many of them eventually get married with their family's blessing. The queen of England is even played by a Black woman!
In that series, everyone is royal (except the servants, who are minor characters in this series), So all kinds of royal characters, whether good, bad, slightly eccentric, etc. are played by all races, which is of no more importance than the color of someone's eyes. Once you get used to it, you don't even notice it.
In the show Hamilton, they do a little of that also, though I did feel that making Thomas Jefferson a Black man was a bit much. (All the other main characters are White or Asian.) I don't remember if anyone ever mentions the fact that Jefferson had many slaves, and one of them was his mistress, and possibly had children with him.
To get back to the original question, I still do think that including that silly jingle was unnecessary and had no importance to the plot, so why not just cut it? Movies are edited all the time, mainly for length reasons, but sometimes because a scene is too horrible or something like that. If the Eeny Meeney thing had some kind of dramatic significance, I could be wrong about wishing they had removed it. But I don't see it. Maybe I'll watch it again and get some new enlightenment. I do know I would have enjoyed the movie a lot more without it.