What a great topic!
I have many ear worms, which often come visit me.
Like the song "Mrs. Robinson" or Bob Marleys "I shot the Sheriff" and some German melodies. In particular those of childhood.
I compare my ear-sense with the smell-sense and the touch-sense and call it "body memory" - the strongest seem to have built up in my early years. So yes, I have made the same experiences you describe.

Once one has learned to ride a bike or to swim, this can hardly be forgotten again.

"The piper's maggot'" is a nice term but I will stick with ear-worm as this is my native tongue:)

The chain of associations you gave makes absolut sense- and shows that what I hear, touch, smell and taste is stronger when it is related to my outer environment. I also have a strong memory of Bachs composition "Toccata and Fugue" played on the Organ - awesome!!!
Don't listen to it, it will give you the greatest worm ever! :-)) - my very long relationship with a DJ awakens memories of him playing this peace on my electronic organ - a Yahama - which cost my parents around 3.000 Deutsche Mark back then.

people tend to remember unfinished tasks better than completed ones
I can confirm that for myself.

I think it has to do with my not making it into the mastery of composition of any kind related to music. No milage in practicing, no completed tasks.

In order to get rid of it: I would say the best way is to totally accept that they are there. Once the teeth are unclenched and mind relaxed, they'll disappear as they came.

"Zeigarnik" is just an insane name!! LOL!!