When it comes to the popularizers, they do go overboard in describing things that are more philosophical that physical. However while the MWI is indeed “philosophy of physics”, many would think that it is unfair to draw extra attention to this fact just because it seems bizarre to you. There are many, many bits of ontology that are technically speaking “philosophy” scattered throughout our physics education, and even built into our theories. Most of them you wouldn’t question because they are “obvious” to you, so you are applying a double-standard here. And these bits of philosophy are valuable! It’s very useful to be able to visualize things, and have some underlying picture of reality to fall back on when trying to work out the solution to a problem. As you know from your description of Einstein’s work, a physicist’s intuition about the nature of reality has shown time and time again to be invaluable in moving science forward. An while this intuition is guided by what is technically philosophy, it is in practice very much a part of pragmatic physics. And in moving physics forward, it helps to be able to try to understand “what is really going on.” How else are we to make that “leap” to the next big theory? But when you have mathematics that has many possible interpretations as to “what is really going on” you have to make a mental choice in order to move forward. Typically “beauty” is involved, and of course Occam’s razor. This is, whether you like it or not, a valuable part of practicing physics that is technically philosophy.
To sum up the above paragraph, it is unfair to characterize MWI as philosophy (not that there is anything wrong with philosophy) unless you also characterize the copenhagen interpretation as equally philosophy. If you are going to learn quantum mechanics, it is normal (unless you are an android or are autistic or something) to adopt some mental description of what is actually happening, something that corresponds to the math and is able to give you physical intuition. You have to make some choice. That choice is philosophy, whether you end up on the side of the MWI, copenhagen, or anything else. So it is unfair to pick out the MWI as somehow a special offender. Your real issue with it is that you think it is strange and unnecessarily extravagant. In my opinion this is because you do not understand it well enough.
It offers no useful method. It is less ugly because it makes fewer assumptions. The best I can do is to offer a bit of history and my comment on it.
In the 1920’s it was found that the evolution of particles can be described by a simple differential equation (the Schrodinger equation). This is simple and beautiful. A given particle is described by a wave, which can be many places at once. This wave undulates and evolves according the the Schrodinger equation. But when we make a measurement, we don’t see a wave, we see a ‘blip’. Almost 100 years later today, as far as we can tell, the Schrodinger equation is correct whenever we are not looking. But still, no one knows why or how or what causes the Schrodinger equation to suddenly stop working whenever we make a measurement. Clearly something really crazy is going on! Some complicated physics that causes the wave to “collapse” whenever we try to look at it! But no one has ever figured out any physical interaction that could cause a collapse, because every physical interaction is correctly described by the Schrodinger equation! Except when we are looking! But we are just made up of particles. Our bodies follow the Schrodinger equation too! So there should be no such thing as a special kind of measurement we can make, that would cause a wave function to collapse. How can we cause the wave function to collapse by making a measurement, if we are just made up of particles, and the Schrodinger equation is always correct for particles and doesn’t cause any collapse? Clearly there is a fundamental logical inconsistency in trying to think of the wave function as collapsing. It is a total mess, requires a world-view that is akin to the magical dragons you have mentioned, and some new physical mechanism causing wave function collapse which is ill-defined (what is a measurement?) and self-contradictory. This is why people have tried to use consciousness to explain wave function collapse. This is the ugly mess people get in when they are taught that the wave function actually collapses!
The far simpler explanation is that the Schrodinger equation is always correct. There is no such thing as wave function collapse. There is just the appearance of wave-function collapse due to anthropic self-selection and decoherence. This is a fancy way of saying that there is a large, continuous wave function, and you take that wave function seriously. You don’t postulate “many worlds” so much as you simply assume the wave function exists and evolves according to the Schrodinger equation. The “interpretation” comes in when you notice that the wave function is a sum of many little pieces (like when you integrate a curve you can break it into pieces that you sum up). Because the Schrodinger equation is linear, the evolution of the whole wave function is equivalent to the simultaneous evolution of an infinite number of slightly different “pieces” that make up the wave function. One of those pieces is “you”. Another piece is a slightly different version of “you”. Each version of “you” interacts with pieces of the wave function of the surrounding universe, and becomes entangled with it (interactions mean that conservation laws start forbidding mutually incompatible alternate possibilities). As each version of “you” gets more and more entangled with parts of other waves functions, it becomes impossible for “you v1” to interact with “you v2” because “you v1” has gone and interacted in a way that is logically incompatible with “you v2”. In other words, “you v1” and “you v2” are in “separate universes” practically speaking, although you are both still part of a larger wave function still evolving according to the Schrodinger equation.
It may sound complicated, and it is difficult to explain, and yes, deducing all of this may seem complicated, and some of the deductions may seem bizarre to you, but the underlying idea is very simple and coherent, and not bizarre at all. The universe consists of a wave function that evolves according the Schrodinger equation. Full stop.