Why do we write electrode potentials next to chemical reactions? We might have a half cell consisting of the redox couple $\ce{Ag+}/\ce{Ag}$, which for example's sake might be fixed to be at the cathode (i.e. undergoing reduction).

The reaction at the cathode is then

$$\ce{Ag+ + e- -&gt; Ag}$$

The standard electrode potential of the $\ce{Ag+}/\ce{Ag}$ couple is $+0.80\ \mathrm V$ with respect to SHE. I've seen the above reaction written as such:

$$\ce{Ag+ + e- &lt;=&gt; Ag} \quad E = +0.80\ \mathrm V$$

I'm confused as to why we write this electrode potential next to the reaction, considering that the electrode potential of a redox couple is reasonably disjoint to the concept of the chemical reaction. Although, at the end of the day, the chemical reaction results in the interfacial potential difference which causes the electrode potential, $E$ is a property of the electrode, not the chemical reaction.

Is it perhaps just because we want to write the equation of the reaction occurring at a certain electrode and the electrode potential of the electrode on the same line?

I'm aware that for associating potentials with written reactions, there exist two different conventions: European, where all reactions are labelled with the reduction potentials of the couple, and American, where oxidation reactions are labelled with the oxidation potentials of the redox couple and likewise reductions with reduction potentials. However, my question is why we need to attach the electrode potential to the chemical reaction in the first place! 

That is, ultimately, aren't redox potentials associated with redox couples/electrodes, not reactions?
Your question is a valid question, and ignore downvotes. They don't mean anything.

Your understanding is very good and that you realized that the electrode potential is a property of the electrode and it really does not care how the reaction is written. However, a equation is $needed$ to keep track of the electrons lost or gained in the Nernst equation. So as a tradition and a matter of convenience, electrode potentials are quoted along with a balanced half-cell.

Another very important thing to remember (even some university teachers and PhDs lack this understanding), that the sign of the electrode potential in invariant. It does not care how the reaction is written. It is the electrostatic sign of the electrode. This part takes care of the European and American conventions. American convention is due to Gibbs and Latimer and no longer used. Most international electrochemists have agreed to stick to the European convention i.e., write all half reactions as reduction.