First, although it is common in some textbooks, I don't think it is a good thing to necessarily relate the equiprobability postulate to ergodicity.
Second, what this postulate enables is to estimate the probability distribution for the macrovariable you want to look at. You can of course look at the most probable value for this macrostate and this will correspond to a "thermodynamic interpretation" of what you can expect to observe.
However, in statistical mechanics, what you can expect is not the most probable value but rather the average value and they need not be the same.
Moreover, you may want more than simply the average value; you can also want to predict what is the free energy difference between one value of the macrostate and another in some kind of transformation in your sytem and for this you are in trouble if you don't try to get the probabilities right.
Suppose you have a box of volume $V$ filled with a mol of an ideal gas with internal energy $E$. This defines the macrostate of your system, or intuitively, how your system looks in a macroscopic scale. However, we still don't know how it looks in a microscopic scale, i.e., we don't know how the $\sim 10^{23}$ particles over there are behaving individually. There are many different possibilities, which are the microstates of the system. For example, at time $t = t_0$ they could have positions $x_i$ and velocities $v_i$, where the index runs over all the particles. This is one particular microstate. However, the macrostate would be the same if particle $i=1$ had position $x_2$ and velocity $v_2$ while particle $i=2$ had $x_2$ and $v_2$ (I'm assuming things are classical and indistinguishable for simplicity). So which is the correct microstate?
From a macroscopic point of view, we don't know. All we can do is attribute what is the probability of the system being in each possible microstate. The principle you stated implies that both microstates I exemplified are equally likely to be the actual microstate. We don't know which is the right microstate, and all the possible ones are equally likely.
The system moving towards the largest number of microstates is then not only a change of microstates, but also a change of macrostate. If I mix my gas with another box of gas at different temperature or something, the system will reach equilibrium at the macrostate with the most possible microstates. We still won't know what is the right microstate, being able only to attribute probabilities.
Essentially, as OP pointed out in the comments, the idea is that since we do not know which microstate is the correct one, we assign equal probabilities to all of them.
Now, this does have a bit of nuance. Is it always valid to do this? In fact, it depends on the information you have about your system. Instead of an ideal gas, let us pick a generic gas. If the energy is fixed, then all available microstates should have the very same internal energy and there is no reason to prefer one of them over the other ones. We call this the microcanonical ensemble. On the other hand, suppose temperature (which is related to the expectation value of energy) is fixed. In this situation, there could be states with more internal energy than others, as long as the temperature stays the same (for the ideal gas, this won't happen because the energy is proportional to temperature, but let us consider a more general scenario). In this situation, it can be more likely for microstates with lower internal energy to occur, so we won't pick all probabilities to be the same. Instead, they are given by a Boltzmann distribution. This is known as the canonical ensemble.
The key point is that since we do not know what is the true microstate, we can only assign probabilities. We do this according to the information we have (or according to the experimental conditions, if you prefer). For fixed energy, all microstates should have the very same probability of being the true microstate, so they are, in this sense, equally likely.
Best Answer
The "search" for lowest energy states is a transient effect. If a system is in a high energy state and can emit energy out of the system, it will do so until the energy of the system diminishes to where it is no longer energetically favorable to do so.
This "isolated system" deals with the steady state case, where energy is not entering or leaving for an arbitrarily long time. Steady state systems exhibit different behaviors than transients.
Given two microstates A and B, we know that the probability of a A->B transition occurring must be the same as the probability of B->A because both states have equal energy. It's easy to see that if the probability of those transitions are equal, then the transient case would have smoothed them out before we arrived at steady state. If P(A) < P(B), we would expect more B->A transitions than A->B transitions, by Bayes' theorem, until eventually we reach P(A) = P(B).
Worth noting however: postulates are not provable. It's called the fundamental postulate of statistical mechanics because we can't actually prove that systems will do this. However, all systems yet observed do indeed demonstrate this behavior, even systems that were designed to break it. You could, however, turn it around and say that if the micro-states are not equiprobable, then you by definition do not fully know the composition of the system.