I think you might try approaching this in the Heisenberg picture.
The time derivative of the position operator is:
$$\dfrac{d \hat x}{dt} = \dfrac{i}{\hbar}[\hat H, \hat x]$$
which is a reasonable velocity operator. The time derivative of the velocity operator is then:
$$\dfrac{d^2 \hat x}{dt^2} = \dfrac{i}{\hbar}[\hat H, \dfrac{d \hat x}{dt}]$$
For example, consider a free particle so that $\hat H = \frac{\hat P^2}{2m}$. The velocity operator would then be $\frac{\hat P}{m}$. This certainly looks reasonable as it is of the form of the classical $\vec v = \frac{\vec p}{m}$ relationship.
But, note that the velocity operator commutes with this Hamiltonian so the commutator in the definition of the acceleration operator is 0. But that is what it must be since we're assuming the Hamiltonian of a free particle which means there is no force acting on it.
Now, consider a particle in a potential so that $\hat H = \frac{\hat P^2}{2m} + \hat U$. The velocity operator, for this system, is then $\frac{\hat P}{m} + \frac{i}{\hbar}[\hat U, \hat x]$.
Assuming the potential is not a function of momentum, the commutator is zero and the velocity operator is the same as for the free particle.
The acceleration operator is then $\dfrac{i}{\hbar}[\hat U, \frac{\hat P}{m}]$.
In the position basis, this operator is just $\frac{-\nabla U(\vec x)}{m} $ which looks like the acceleration of a classical particle of mass m in a potential given by $U(\vec x)$.
Let me first say that I think Tobias Kienzler has done a great job of discussing the intuition behind your question in going from finite to infinite dimensions.
I'll, instead, attempt to address the mathematical content of Jackson's statements. My basic claim will be that
Whether you are working in finite or infinite dimension, writing the Schrodinger equation in a specific basis only involves making definitions.
To see this clearly without having to worry about possible mathematical subtleties, let's first consider
Finite dimension
In this case, we can be certain that there exists an orthnormal basis $\{|n\rangle\}_{n=1, \dots N}$ for the Hilbert space $\mathcal H$. Now for any state $|\psi(t)\rangle$ we define the so-called matrix elements of the state and Hamiltonian as follows:
\begin{align}
\psi_n(t) = \langle n|\psi(t)\rangle, \qquad H_{mn} = \langle m|H|n\rangle
\end{align}
Now take the inner product of both sides of the Schrodinger equation with $\langle m|$, and use linearity of the inner product and derivative to write
\begin{align}
\langle m|\frac{d}{dt}|\psi(t)\rangle=\frac{d}{dt}\langle m|\psi(t)\rangle=\frac{d\psi_m}{dt}(t)
\end{align}
The fact that our basis is orthonormal tells us that we have the resolution of the indentity
\begin{align}
I = \sum_{m=1}^N|m\rangle\langle m|
\end{align}
So that after taking the inner product with $\langle m|$, the write hand side of Schrodinger's equation can be written as follows:
\begin{align}
\langle m|H|\psi(t)\rangle
= \sum_{m=1}^N\langle n|H|m\rangle\langle m|\psi(t)\rangle
= \sum_{m=1}^N H_{nm}\psi_m(t)
\end{align}
Equating putting this all together gives the Schrodinger equation in the $\{|n\rangle\}$ basis;
\begin{align}
\frac{d\psi_n}{dt}(t) = \sum_{m=1}^NH_{nm}\psi_m(t)
\end{align}
Infinite dimension
With an infinite number of dimensions, we can choose to write the Schrodinger equation either in a discrete (countable) basis for the Hilbert space $\mathcal H$, which always exists by the way since quantum mechanical Hilbert spaces all possess a countable, orthonormal basis, or we can choose a continuous "basis" like the position "basis" in which to write the equation. I put basis in quotes here because the position space wavefunctions are not actually elements of the Hilbert space since they are not square-integrable functions.
In the case of a countable orthonormal basis, the computation performed above for writing the Schodinger equation in a basis follows through in precisely the same way with the replacement of $N$ with $\infty$ everywhere.
In the case of the "basis" $\{|x\rangle\rangle_{x\in\mathbb R}$, the computation above carries through almost in the exact same way (as your question essentially shows), except the definitions we made in the beginning change slightly. In particular, we define functions $\psi:\mathbb R^2\to\mathbb C$ and $h:\mathbb R^2\to\mathbb C$ by
\begin{align}
\psi(x,t) = \langle x|\psi(t)\rangle, \qquad h(x,x') = \langle x|H|x'\rangle
\end{align}
Then the position space representation of the Schrodinger equation follows by taking the inner product of both sides of the equation with $\langle x|$ and using the resolution of the identity
\begin{align}
I = \int_{-\infty}^\infty dx'\, |x'\rangle\langle x'|
\end{align}
The only real mathematical subtleties you have to worry about in this case are exactly what sorts of objects the symbols $|x\rangle$ represent (since they are not in the Hilbert space) and in what sense one can write a resolution of the identity for such objects. But once you have taken care of these issues, the conversion of the Schrodinger equation into its expression in a particular "representation" is just a matter of making the appropriate definitions.
Best Answer
The rigorous way is to use $e^{i\phi}=c+is$ and the conjugate angular momentum operators, as the Hilbert space is spanned by single-valued functions on the circle. But physicists often get away with using $\phi$ but then remembering that physical results have to be $2\pi$ periodic in $\phi$.