electromagnetism - How does gauge-fixing really work?

Leaving technical issues like Gribov copies and residual gauge freedom aside, how do gauge fixing conditions like the Coulomb condition

$$\begin{equation} \partial_i A_i =0 \end{equation}$$ or the axial condition $$\begin{equation} A_3 =0 \end{equation}$$ help in getting rid of the gauge redundancy?

A first idea is that conditions like this help us to fix the gauge function $\eta(x_\mu)$. Before any gauge fixing takes place, we have the freedom to use different configurations $A_\mu$ to describe the same physical situation that are related by

$$A_\mu (x_\mu) \to A'_\mu \equiv A_\mu (x_\mu) + \partial_\mu \eta(x_\mu ).$$ If we plug $A'_\mu$ into a gauge condition, like the Coulomb condition, we can derive an for the gauge function $\eta(x_\mu$): $$\begin{align} 0 &=\partial_i A'_i \\ &= \partial_i A_i (x_\mu) + \partial_i \eta(x_\mu ) \\ \to \partial_i \eta(x_\mu ) &= -\partial_i A_i (x_\mu) \tag{1} \end{align}$$ If we now use a specific solution of the equation of motion $A_\mu$, we can solve this equation to find a specific gauge function $\eta(x_\mu )$. But why is this useful at all if there is nothing that specifies what $A_\mu$, we should put on the right-hand side of Eq. 1?

In more concrete terms, let's say we have a solution of the equation of motion $A_\mu$ and another configuration $A'_\mu$ that is related to $A_\mu$ by a gauge transformation. My problem is that even if we choose one specific gauge function $\eta$, we still have no idea whether we should use $A_\mu (x_\mu) + \partial_\mu \eta(x_\mu )$ or $A'_\mu (x_\mu) + \partial_\mu \eta(x_\mu )$ and thus the gauge redundancy is still there. After all, there is nothing that tells us that $A'_\mu$ isn't the "original" solution of the equation of motion and $A_\mu$ just a gauge transformed version of it.

Answer

Leaving technical issues like Gribov copies and residual gauge freedom aside, how do gauge fixing conditions like the Coulomb condition $∂_𝑖 𝐴_𝑖=0$ or the axial condition $𝐴_3=0$ help in getting rid of the gauge redundancy?

Because while there are a huge number of fields $A_\mu$ that satisfy the defining equation $\partial_\mu A_\nu - \partial_\nu A_\mu = F_{\mu \nu}$ (where $F_{\mu \nu}$ is uniquely physically determined), there are a much smaller (though still infinite) number of fields $A_\mu$ that satisfy that equation and also solve the partial gauge fixing equations that you list above. Sometimes this narrowing down is useful, sometimes it isn't - it depends on the situation. For example, the covariant version of the two sourced Maxwell's equations takes on a simpler mathematical form in Lorentz gauge than in an arbitrary gauge. But these partial gauge fixing conditions don't fully pin down a unique $A_\mu$ field by themselves.

A first idea is that conditions like this help us to fix the gauge function $\eta(x_\mu)$. Before any gauge fixing takes place, we have the freedom to use different configurations $A_\mu$ to describe the same physical situation that are related by

$$A_\mu (x_\mu) \to A'_\mu \equiv A_\mu (x_\mu) + \partial_\mu \eta(x_\mu ).$$ If we plug $A'_\mu$ into a gauge condition, like the Coulomb condition, we can derive an for the gauge function $\eta(x_\mu$): $$\begin{align} 0 &=\partial_i A'_i \\ &= \partial_i A_i (x_\mu) + \partial_i \eta(x_\mu ) \\ \to \partial_i \eta(x_\mu ) &= -\partial_i A_i (x_\mu) \tag{1} \end{align}$$ If we now use a specific solution of the equation of motion $A_\mu$, we can solve this equation to find a specific gauge function $\eta(x_\mu )$. But why is this useful at all if there is nothing that specifies what $A_\mu$, we should put on the right-hand side of Eq. 1?

You're right, there is no unique transition function $\eta$ that takes you into (say) Coulomb gauge - it depends on whichever $A_\mu$ you happened to start with. How you get to Coulomb gauge depends on where you're starting from, just as how to get to Chicago depends on whether you're starting from Boston or Seattle. In practice, you just start with a totally random choice of $A_\mu$, then solve the equation you give above to find an appropriate $\eta$, then add the gradient of that $\eta$ to your original $A_\mu$ to form $A_\mu'$, then you can completely forget about your original $A_\mu$ and $\eta$.

In more concrete terms, let's say we have a solution of the equation of motion $A_\mu$ and another configuration $A'_\mu$ that is related to $A_\mu$ by a gauge transformation. My problem is that even if we choose one specific gauge function $\eta$, we still have no idea whether we should use $A_\mu (x_\mu) + \partial_\mu \eta(x_\mu )$ or $A'_\mu (x_\mu) + \partial_\mu \eta(x_\mu )$ and thus the gauge redundancy is still there. After all, there is nothing that tells us that $A'_\mu$ isn't the "original" solution of the equation of motion and $A_\mu$ just a gauge transformed version of it.

You don't "choose one specific [transition] function $\eta$" ahead of time in order to gauge-fix. The choice of transition function inherently depends on your (arbitrary) starting field and so itself is pretty arbitrary.