Are there any proofs that use $0/0$ is indeterminate? I'd imagine that "$n/0$ is undefined $\forall n\neq 0$" is very useful in finding contradictions, but are there any proofs that somehow use "$0/0$ is indeterminate"?
Yes. You can use that $0/0$ is indeterminate to conclude that certain expressions are in fact $0$ (and this is pretty much the only thing you can use $0/0$ for). However, nobody writes down such arguments because:


Writing such an argument requires you to be very careful, hence requires a lot of effort.
Any such argument can be replaced by a much easier argument of the form "Either the expression is $0$, or I get a contradiction: therefore the expression is $0$".


For an example of how to use that $\dfrac 00$ is indeterminate (and for a more general explanation of what it means to "divide by $0$", see this answer). On the flip-side, the fact that no number corresponds to the faction $\dfrac n0$ for $n\neq0$ is incredibly useful: for example, if you have an expression $\lim\dfrac {f(x)}{g(x)}$ and you (carefully) reduce it to an expression of the form $\dfrac n0$ where $n\neq0$, you can conclude that the limit does not exist, without mucking about with $\epsilon$'s and $\delta$'s.
Hope I understood you correctly, but here's a very simple example.

consider $\lim_{x \to 0} \frac{x}{x}$. it tends to one.

Now consider $\lim_{x \to 0} \frac{2x}{x}$. it tends to two.

We have 2 different limits to something that is close to $\frac{0}{0}$. 

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