I have some examples, but I might have more words of warning than examples. Example 1. Let's say you have just the axioms for the real numbers, and you want to establish that $2$ is not zero, but all you know is that $1$ is not zero (perhaps because you want to divide by 2 for whatever reason). Knowing that there is a field with $2$ elements where $2= 0$, you discover you have no choice but to prove 2 is positive, even though it's not what you were going for. (This is an example where taking on a more general point of view does not give you something different to prove instead, but limits the approaches to what you're trying to prove.)

So in order to prove 2 is positive, it suffices to prove 1 is positive. And here it somehow just makes sense to prove that the square of any nonzero number is positive -- from this point of view 1 is positive "because it's a square".

Here's another example from real analysis. You want to prove things about the cube root function from first principles. Like the fact that it's well-defined, continuous, differentiable, concave, etc. Typically one approaches by proving general theorems about smooth, increasing functions, and applies these general theorems to the function x^3 to learn about its inverse.

So while this seems like the kind of example you're going for, I personally don't like the example only because it suggests this is the "right" way to go. Often when one takes a "generalize" approach solving the problem can become easier, especially if you already know the general facts which end up doing the job. But for this example, defining the cube root is just a bit easier than using the general intermediate value theorem since you can define it as $f(x) = \sup \{ t : t^3 \leq x \}$ and prove directly that $f(x)^3 = x$.

Moreover, it's also a good example to use the implicit definition to prove things directly for this function just because it makes manipulations concrete. For example, since $f(x+h)^3 = x + h$, we have

$ (f(x+h)^3 - f(x)^3) = h $

is also equal to

$ (f(x+h) - f(x)) \cdot \int_0^1 3 (s f(x+h) + (1-s) f(x))^2 ds $

If you look at this expression for a little while, you will be able to deduce things like monotonicity, uniform continuity on any interval $[\epsilon, \infty)$, differentiability on such intervals, concavity, so on. The point is that even when general technology works, it can be quite instructive to use "general methods", but execute them on explicit examples.

You might argue that these direct proofs are the most insightful, but one is more likely to first find a general proof (especially if the appropriate general apparatus already exists), and then the direct proofs can only be found later indeed because they require a more direct understanding of the specific problem. I personally use the cube root of 7 (which nobody can name) in calculus lectures to motivate the concept of continuity, so I agree here it's a very natural use of the concept to define the cube root.

Differential equations have, I think, a lot of examples. If you want to learn something about the exponential function and your point of view is that it is the unique solution to $\frac{df}{dx} = f$ (or $\frac{df}{dx} = i f )$ with data $f(0) = 1$, then in order to give a differential equation-type analysis to prove the basic properties (which is fun) you will often have to consider the same differential equation with more general data (and in particular prove uniqueness of the $0$ solution). In fact, you may be tempted to consider more general ODE's like $\frac{df}{dx} = F(x, f)$ because this point of view can inspire some of the techniques. It also helps greatly (say if you want to prove $e^{it}$ is periodic) to just know what a vector field is and have that point of view.

I think the problem is that the general theorems can easily end up being forced to consider cases which are more complicated than the problem at hand. Even if you find a proof, that doesn't necessarily mean you should settle for it. If you look at the function $f(x) = 1/k$ where $k$ is the smallest positive integer such that $kx \in {\mathbb Z}$, and you want to show $f$ is Riemann integrable, you can of course prove a theorem that characterizes Riemann integrable functions in terms of the measure of their sets of discontinuity. Of course, if you know this theorem off the top of your head, it is "easy" to prove $f$ is Riemann integrable. But this proof is inefficient because it invokes a theorem that takes care of the nastiest examples of Riemann integrable functions (which this $f$ is not).

One more example: it's very easy to find a faulty proof of the chain rule if your point of view is not general enough. The first thing many people try to do to analyze the difference quotients for $f(g(x))$ is to multiply and divide by $g(x+h) - g(x)$. But then you get into these hairy problems where the number by which you divide may be zero. If you want to find the correct proof, you should take the point of view that the chain rule is a statement which should be true for maps between Euclidean spaces of any dimension. It just says the linearization of the composition is the composition of the linearizations. With this point of view in mind, you should not dare to try dividing, because dividing by $g(x+h) - g(x)$ does not even make sense in this generality.

Here's another: prove that $\int_{\mathbb R} e^{i \xi x} e^{- x^4 } dx$ is bounded by $\frac{C}{(1 + |\xi|)^2}$. Somehow you have to recognize the key features of $e^{-x^4}$ are that it will cancel against something very oscillatory because it is so smooth. Similar example problem: prove the decay for large $\xi$ of $\int_{\mathbb R} \log(1 + .5 \sin(\xi x) ) e^{- x^4 } dx$. It seems like this "look for a more general setting" trick really needs to be drilled into you for it to work because... it may require some imagination or experience to know what the right general setting is.

What some examples show is that finding the proof may be much more difficult, maybe impossible, without a willingness to work in this direction.