> The original Elo system is based on a normal distribution instead of a logistic distribution
>
> ...
>
> If this still works, then why would they have switched to the logistic function?

There hasn't been a switch. The FIDE uses Elo's original system for which a table is computed.  See the [Fide handbook](https://handbook.fide.com/chapter/B022017).

The win-probability (or score,since it includes draws), is computed as 

$$p(\text{A win from B})= \Phi\left( \frac{\text{Elo}_A - \text{Elo}_B}{200 \cdot \sqrt{2}} \right)$$ 

For example, a $100$ Elo difference will relate to a $0.6381632$ win probability for the higher valued player. Or $0.64$ as it is in the tables of the FIDE.

The use of the logistic function/distribution is a useful trick to make an estimate of the values in the table. Also, several other competitions may be using the logistic function. There is not really a large difference between the two and it is just a practical trick to make computations easy.

> Would the Elo system still work if we chose a simpler function like this?
>
> $$E_A = \frac{R_A}{R_A + R_B} = \frac1{1 + \frac{R_B}{R_A}}$$

The system is arbitrary, so yes this would work as well. However, the computations with a linear Elo scale might be easier. 

 - For instance, one could compute an average Elo rating for a pool of chess players and approximated the win probability of a player based on the difference with the average Elo. 
 - Another example is that differences in Elo rating are more easier to
   evaluate. If some top player has rating 13102 and another has a
   rating 4996, than it is a bit awkward to compute the difference in
   the level between the players because this involves a division
   instead of a subtraction.
 - Also, the updating works as a zero sum game. Whenever a player increases by some value $x$, then another player decreases by a value $x$. If you would do this for an exponentially increasing score, then the people at the top will be changing their scores only very slowly (in some sense the system does have a way to reduce the speed of changes in scores for high level players, but it does not reduce the speed *that* much as what would be the case with an exponentially increasing score)