According to my former question "The gradient on sphere", the gradient of a function $f$ in spherical coordinates is given by \begin{align*} \nabla_{\mathbb{R}^3} f &= \frac{\partial f}{\partial r} \frac{\partial}{\partial r} + \nabla_{\mathbb{S}^2(r)} f, \end{align*} and the gradient on the $r$-sphere is $$ \nabla_{\mathbb{S}^2(r)} f = \frac{1}{r^2} \frac{\partial f}{\partial \varphi} \frac{\partial}{\partial \varphi} + \frac{1}{r^2 \sin \varphi} \frac{\partial f}{\partial \theta} \frac{\partial}{\partial \theta}. $$
However, according to the post "What is the metric tensor on the n-sphere (hypersphere)?", the metric of $\mathbb{S}^2(r)$ is given by $$ g_{\mathbb{S}^2(r)} = r^2 d\varphi \otimes d\varphi + r^2 \sin^2 \varphi \, d\theta \otimes d\theta. $$ Using this, we can calculate as follows: \begin{align*} \nabla_{\mathbb{S}^2(r)} f &= g^{11}_{\mathbb{S}^2(r)} \frac{\partial f}{\partial \varphi} \frac{\partial}{\partial \varphi} + g^{22}_{\mathbb{S}^2(r)} \frac{\partial f}{\partial \theta} \frac{\partial}{\partial \theta} \\ &= \frac{1}{r^2} \frac{\partial f}{\partial \varphi} \frac{\partial}{\partial \varphi} + \frac{1}{r^2 \sin^2 \varphi} \frac{\partial f}{\partial \theta} \frac{\partial}{\partial \theta}. \end{align*} I'm confused with the difference between the power of $\sin \varphi$. What went wrong?
Here is one more question. Can I write $\nabla_{\mathbb{S}^2(r)}f = \frac{1}{r} \nabla_{\mathbb{S}^2(1)} f$ if $f : \mathbb{R}^3 \setminus \lbrace 0 \rbrace \rightarrow \mathbb{R}$ satisfies $f(x) = f(x/ \vert x \vert)$?
The point is that you've mistranscribed the result from the first link. There the results are written in terms of unit basis vectors for the tangent plane of the sphere, not $\partial/\partial\phi$ and $\partial/\partial\theta$.