The basic strategy you give is correct: To first approximation, algebraic geometry over $\overline{\mathbb F_p}$ is like algebraic geometry over $\mathbb C$ except for extra phenomena. Usually, I think of these extra phenomena as "inseparable maps" and "wild ramification".

Inseparable maps include Frobenius, but not every inseparable map is a Frobenius. They all have properties which are unexpected from a characteristic zero point of view (e.g. rank of the derivative of the map a a generic point does not equal the dimension of the image).

Wildly ramified covers also behave differently from characteristic zero maps, but in a more subtle way. Perhaps the biggest is that to specify a finite branched cover in characteristic zero it suffices up to only a finite ambiguity to specify the branch locus, while in characteristic $p$ the ambiguity can be infinite.

Of course as one gets more experience in algebraic geometry in characteristic $p$ one increasingly thinks about characteristic $p$ problems by reference to previous characteristic $p$ situations one has encountered, but I think "characteristic zero plus inseparable maps and wild ramification" is a good starting point.

In terms of visualizing, the only reasons to divert from visualizing algebraic curves as complex surfaces and so on are the same reasons that exist in characteristic $0$: sometimes, to fit more in your visual imagination, you want to visualize algebraic curves as curves, algebraic surfaces as surfaces, and so on, and sometimes you want to visualize arbitrarily-high-dimensional varieties, which means you have to imagine them as two- or maybe three-dimensional.

Let me also briefly answer your formal question in the negative: The set of maps $\mathbb P^1_{\mathbb C} \to \mathbb P^1_{\mathbb C}$ is uncountably large, and adding extra morphisms couldn't make this countable, but the set of maps $\mathbb P^1_{\overline{\mathbb F_p} } \to \mathbb P^1_{\overline{\mathbb F_p}$ is countable.

The basic strategy you give is correct: To first approximation, algebraic geometry over $\overline{\mathbb F_p}$ is like algebraic geometry over $\mathbb C$ except for extra phenomena. Usually, I think of these extra phenomena as "inseparable maps" and "wild ramification".

Inseparable maps include Frobenius, but not every inseparable map is a Frobenius. They all have properties which are unexpected from a characteristic zero point of view (e.g. rank of the derivative of the map a a generic point does not equal the dimension of the image).

Wildly ramified covers also behave differently from characteristic zero maps, but in a more subtle way. Perhaps the biggest is that to specify a finite branched cover in characteristic zero it suffices up to only a finite ambiguity to specify the branch locus, while in characteristic $p$ the ambiguity can be infinite.

Of course as one gets more experience in algebraic geometry in characteristic $p$ one increasingly thinks about characteristic $p$ problems by reference to previous characteristic $p$ situations one has encountered, but I think "characteristic zero plus inseparable maps and wild ramification" is a good starting point.

In terms of visualizing, the only reasons to divert from visualizing algebraic curves as complex surfaces and so on are the same reasons that exist in characteristic $0$: sometimes, to fit more in your visual imagination, you want to visualize algebraic curves as curves, algebraic surfaces as surfaces, and so on, and sometimes you want to visualize arbitrarily-high-dimensional varieties, which means you have to imagine them as two- or maybe three-dimensional.

Let me also briefly answer your formal question in the negative: The set of maps $\mathbb P^1_{\mathbb C}\to\mathbb P^1_{\mathbb C}$ is uncountably large, and adding extra morphisms couldn't make this countable, but the set of maps $\mathbb P^1_{\overline{\mathbb F_p}}\to\mathbb P^1_{\overline{\mathbb F_p}}$ is countable.