The short answer is that the visualizations of finite fields are different from what you seem to want. I will elaborate a bit, though. It is not clear that anything I suggest is really a "visualization" or a "geometric point of view". But they do provide a way of getting your hands on a finite field starting from structures inside the complex numbers.
Let's first look at the prime field $F_p=\mathbf{Z}/p\mathbf{Z}$. As you observed, we can faithfully map the additive structure of $F_p$ to the unit circle via the (obviously well-defined) map $ e:m+p\mathbf{Z}\mapsto e^{2m\pi i/p}. $ This is a homomorphism from the additive group of $F_p$ to the multiplicative group $\mathbf{C}^*$ of non-zero complex numbers, or $ e(x+y)=e(x)e(y)$ for all $x,y\in F_p$. In other words $e$ is what is called an additive character of $F_p$. As you observed, it doesn't really reflect the multiplicative structure of $F_p$ at all. This was to be expected, as we used the multiplicative structure of $\mathbf{C}^*$ to represent the additive structure of $F_p$, so naturality was broken down. Characters and character sums do play a role in the study of number theoretic properties of finite fields.
If we want to take a geometric viewpoint of the prime field $F_p$, $p$ a prime, then I would stick to the number line $\mathbf{R}$. Except that I really need restrict to the integers $\mathbf{Z}$, and then go to the quotient ring, or equivalently identify the integers within a coset of $p\mathbf{Z}$.
You can identify some finite fields by similarly identifying cosets of the ideal $p\mathbf{Z}[i]$, where $\mathbf{Z}[i]=\{a+bi\mid a,b\in\mathbf{Z}\}.$ Geometrically you identify here points $a+bi$ and $a'+b'i$, iff $p\mid (a-a')$ and $p\mid (b-b')$, or instead of "wrapping the line around a circle" you "wrap the plane around a torus". An immediate word of warning. This only works, if $p\equiv-1\pmod4$. So you can construct $F_9$ in this wasy, but you cannot construct $F_4$ or $F_{25}$. One of the reasons why this construction does not work in those cases is that while the resulting quotient ring has the desired number of elements, it is not a field because it has zero divisors. For example, $(1+i)^2=2i\equiv 0 \pmod{2\mathbf{Z}[i]}$ in the ring $\mathbf{Z}[i]/2\mathbf{Z}[i]$, and $(2+i)(2-i)=5\equiv 0$ in the ring $\mathbf{Z}[i]/5\mathbf{Z}[i]$ (I plead guilty to the standard abuse of identifying a number with the coset it represents, so I really meant to write that the square of the coset $(1+i)+2\mathbf{Z}[i]$ is equal to zero in the first example above.
To get fields like $F_4$ and $F_{25}$ we need quotient rings of other lattices of complex numbers. Let $\omega=e^{2\pi i/3}=(-1+i\sqrt3)/2$. Then $\mathbf{Z}[\omega]=\{a+b\omega\mid a,b\in\mathbf{Z}\}$ is a ring (a subring of the complex numbers), and we get (requires some checking) $ F_4=\mathbf{Z}[\omega]/2\mathbf{Z}[\omega]\qquad\text{and}\qquad F_{25}=\mathbf{Z}[\omega]/5\mathbf{Z}[\omega]. $ But a construction for all the finite fields (let alone the algebraic closure) in this way is a bit trickier. The field $F_{p^n}$ is an $n$-dimensional space over $F_p$. A lattice in the complex plane is naturally (at most) a two-dimensional thing, so when $n>2$ it gets dirty. No more wrapping this nicely. It can be done by using certain subrings of $\mathbf{C}$ namely rings of integers of an algebraic number field, but some algebraic number theory is needed. See this question for a description.
We can think of a finite field using roots of unity. But the (non-zero) elements of $F_q$, $q=p^n$, $p$ a prime, are roots of unity of order that is a factor of $q-1$. So, if we want to represent ther multiplicative structure, we need to use those roots of unity. For example, we can also view $F_7$ as the quotient ring $\mathbf{Z}[\omega]/(3+\omega)\mathbf{Z}[\omega]$. When we do that the non-zero elements of $F_7$ are represented by the cosets of the sixth roots of unity, i.e. the cosets $(-\omega)^j+(3+\omega)\mathbf{Z}[\omega]$. This is similar in spirit to the discussion in the linked question.
Things become more uniform, when instead of using the complex numbers we use the $p$-adic integers $\mathbf{Z}_p$ (not to be confused with the residue class ring $\mathbf{Z}/p\mathbf{Z}$. If $q=p^n$, and $\zeta$ is a root of unity of order $q-1$, then we have an isomorphism $ F_q=\mathbf{Z}_p[\zeta]/p\mathbf{Z}_p[\zeta] $ between the finite field $F_q$ and the quotient ring of an extension ring of the $p$-adic integers. That's a rather different animal, and this last paragraph may be meaningless to you unless you have the right background. The $p$-adic integeres themselves cannot be readily visualized geometrically, because the metric there is very weird.