The MWI is what you get if you take seriously the idea that the equations of motion of quantum theory describe reality.
In classical physics the evolution of a physical quantity, such as the $x$ position of a particle is described by a variable $x(t)$ and if you measure the $x$ position you get the value $x(t)$.
In quantum physics the equations of motion of a physical quantity like the $x$ position are written in terms of a Hermitian operator called an observable $\hat{X}(t)$. The possible results of measuring the $x$ position are given by the eigenvalues of $\hat{X}(t)$ and quantum physics predicts the expectation value $\langle\hat{X}(t)\rangle$ using the relative state $\rho$ which is a Hermitian operator with unit trace according to the Born rule:
$$\langle\hat{X}(t)\rangle=\operatorname{tr}(\rho\hat{X}(t))$$
In quantum theory in general the expectation value of an observable at the end of an experiment depends on what happens to all of the possible values of an observable during the experiment: this is called interference. For an example see Section 2 of this paper
https://arxiv.org/abs/math/9911150
If information is copied out of a system this suppresses interference: an effect called decoherence
https://arxiv.org/abs/1911.06282
For systems you see in everyday life, information is copied out of them on a timescale much shorter than the timescale over which they change significantly so interference is very heavily suppressed. As a result, there are multiple versions of those systems with different values of the monitored observables that evolve autonomously and they form layers each of which evolves approximately like the universe as described by classical physics:
https://arxiv.org/abs/1111.2189
https://arxiv.org/abs/quant-ph/0104033
If you measure the $\hat{Z}$ observable of a system $S$ in the state
$$a|\uparrow_z\rangle_S+b|\downarrow_z\rangle_S$$
onto a measuring device $M$ the resulting state looks like this
$$a|\uparrow_z\rangle_S|\uparrow_z\rangle_M+b|\downarrow_z\rangle_S|\downarrow_z\rangle_M$$
One question to ask is how do you expect probabilities to act in reality. You can then look at what quantities actually act as you expect probabilities to act. The probabilities have to respect what information it's possible for you to get out of the system and it has to respect the symmetries of the resulting state. One approach to explaining the Born rule along such lines is called envariance:
https://arxiv.org/abs/quant-ph/0405161
There are other restrictions you can place on probabilities, like a later measurement can't change the probabilities of results of an earlier measurement. For example, if you measure the state
$$\tfrac{1}{\sqrt{3}}|\uparrow_z\rangle_S+\sqrt{\tfrac{2}{3}}|\downarrow_z\rangle_S$$
the equal probability rule you propose would lead you to say the probability of $\uparrow_z$ is 1/2. But suppose you follow up this measurement by saying that if you get the $\downarrow_z$ you will measure the $\hat{X}$ observable which is unsharp when $\hat{Z}$ is sharp so that now there are three outcomes of the resulting overall measurement. Then the equal probability rule would say the probability of $\uparrow_z$ is 1/3 not 1/2. There is an approach to probability in the MWI where you ask how you could assign probabilities to stop inconsistencies like that from being used to construct a Dutch Book against you and that leads to the Born rule, see
https://arxiv.org/abs/0906.2718
In either case working out probabilities involves using more structure than just the number of worlds.
