Really don't like that statement of wikipedia. Of course in the middle, they can be fitted to be quite similar, but they behave differently quite soon at the edges.

Even the math discussions devolve into bickering about edge cases

1) If you don't care about the edge cases, it's not math.

2) Bayesian statistics is not an edge case, but the example explicitly given in the wiki article.

Note of course I'm not talking about rating systems here, I have no idea and not thought about what distribution would fit better in this context, if WW says it's logistic I would just believe it. I don't see any reason why it should be Gaussian, as I don't think you are in a regime for Central Limit Theorem here.

Edit: What come's next is not really well thought off:

If the remainings of my understanding of TS is right, I somehow think the model lacks a parameter anyway. You have mean skill, ok, and the uncertainity of the system on you skill. Somehow it's maybe reasonable to assume Gaussian on the uncertainity. But the same distribution is also used to get the winprobabilities given mean and uncertainity (or?), and I don't see any reason why one should do that.