The collapse of the wave function upon measurement is the interpretation part that is disputed by other interpretations.
In many worlds, collapse is a subjective and approximate phenomena, not something fundamental.
That's perhaps the Narcissistic Many Worlds interpretation. Another way of thinking about Many Worlds is that there's only one universe, in which: after measuring a particle in a superposition of two states, you are now in a superposition of two states. Each of the superimposed you states thinks your instrument measured a single clear result.
The wavefunction of the universe goes on propagating as usual, including both of the superimposed yous, but within each of the superimposed states, that you thinks that it has witnessed wavefunction collapse.
The measurement results are mathematically necessary. The Copenhagen interpretation, that this means there is a physical collapse in the wavefunction, is not.
> To me, it seems that other theories are just disputing what the "collapse" fundamentally means.
If you substitute "collapse" for "measurement" this is pretty much true. What else could "different interpretations of a theory" mean?
One is about what actually happens. This contains Shrodinger equation and similar things.
The other is about what results will we get if we poke the particles with macroscopic objects disturbing them beyond recognition. That's the all math where the word "measurement" is used.
Somehow we think the science of what happens to a frog when you poke it with a knife is a part of zoology. It's important, it might be even more important than zoology, but that's a different domain of science.
The way I see it, other interpretations have no purpose except to reconcile what we observe on small scales with incorrect statements such as "electron is a tiny ball", or "electron is a wave in the three-dimensional space", or "an observer which are themselves composed of quantum mechanical particles is independent of the system, and exerts free will to choose measurements while the observed system is purely deterministic".