So firstly, let's think about what it means to solve the Schrodinger equation for an atom? So the Schrodinger equation describes the evolution of something called a wave function, usually denoted $\Psi({\bf{x}},t)$. The wave function in essence encapsulates all the information about the system you want to model. So if you were, let's say, trying to model the Helium atom, you'd try to work out what the wave function for a Helium atom is. Now the cool thing about the Schrodinger equation is that it can be understood in a fairly intuitive way; let's take a look:
$$i\hbar\cfrac{\partial}{\partial t}\Psi({\bf{x}},t)=\left(-\cfrac{\hbar^2}{2m}\nabla^2+V(x,t)\right)\Psi({\bf{x}},t)$$
So what does any of that mean? The L.H.S. is specifically referring to how the system in question evolves in time. The R.H.S. tells you $\bf{how}$ the evolution is driven. In fact, the R.H.S. is telling you that the dynamics of your system are governed by the total energy of the system, i.e. the Hamiltonian. Now, the great thing about this equation is that it really talks about measurements. To $\bf{measure}$ the dynamics of your system, you effectively need to measure the total energy of your system. Now for $V=0$, you get the Schrodinger equation for a $\textbf{free particle}$. Now when it was said that 'for a single particle, there exists no closed-form solution', well that's not quite the case. The fact that the Schrodinger equation is a second-order partial differential equation is really just a statement that its solutions involve equations with multiple variables, e.g. space in 3 dimensions and time $\textbf{and}$ the equation involves second derivatives. Let's consider for simplicity, the 1D case:
$$i\hbar\cfrac{\partial}{\partial t}\Psi(x,t)=-\cfrac{\hbar^2}{2m}\cfrac{\partial^2}{\partial x^2}\Psi(x,t)$$
This is really just the wave equation. We know the general solution for the wave equation and it's not too tricky to get, that is, the standard plane wave:
$$\Psi(x,t)=Ae^{i(kx-\omega t)}$$
Plug this into the Schrodinger equation and out pops the de Broglie relations, specifically $E=\hbar\omega$. So there you go, the plane wave is a solution to the Schrodinger equation as long as the following relation is true: $E=\hbar\omega$, which in essence tells you that plane waves really just describe particles that carry a discrete packet of energy, or $\textbf{Quanta}$. Now the problem comes when you try to physically realize this solution. The special thing about the wave function is that taking the square of the modulus gives you a probability distribution, which tells you the probability of finding a particle at some $x$ and $t$; specifically
$$|{\Psi(x,t)}|^2 = Pr(x,t)$$
For this solution to be $\textbf{Physical}$, or shall I say describe reality, we would like the following condition to hold:
$$\int_{-\infty}^{\infty}|\Psi(x,t)|^2dx <\infty $$
This means that we can find some number $A$ such that
$$A\int_{-\infty}^{\infty}|\Psi(x,t)|^2dx =1 $$
This tells us that the probability of finding the particle somewhere in space is 1, e.g. the particle (or system) exists. When it comes to free particles, such a condition is not met, so in general, we can't realize the solution to a free particle as a physical solution, despite such a solution being a perfectly fine solution to the Schrodinger equation (you can do a bit of tinkering to show that the free particle can be realized physically as a discrete wave packet).
Now the potential term in the equation describes some configuration that determines the dynamics of your system. More generally, your system will be described by a Hamiltonian. The most simple example is probably the Hamiltonian which describes a particle trapped in a box/infinite potential well. The potential for this system will impose boundary conditions such that your particle can never overcome the potential energy of the system. The boundary conditions allow you to derive a set of solutions by solving the Schrodinger equation. These solutions will correspond to the discrete energy levels that the particle can occupy. There are of course many other boundary conditions you can impose to try and model different systems; this is one way the potential works. The quantum harmonic oscillator is another very important example. In fact, it has a myriad of incredible properties! For example, it can be used as a building block for more complex systems. In fact, you can show that if you have a system of coupled (joined together) simple harmonic oscillators, the Hamiltonian of such a system is exactly the same as the Hamiltonian for an un-coupled set of simple harmonic oscillators, which is actually a big motivating factor for QFT.
We can also have even more complex systems, for example, the Hydrogen atom. The Hamiltonian which models the Hydrogen atom is one that involves Coulomb's law:
$$E\Psi({\bf{x}},t) = -\cfrac{\hbar^2}{2\mu}\nabla^2\Psi({\bf{x}},t)-\cfrac{q^2}{4\pi\epsilon_0r}\Psi({\bf{x}},t)$$
This is effectively a two-body problem and with some clever manipulation, it's not too hard to arrive at the solutions. We can decompose such a system into two parts: one that describes a free particle, e.g. the dynamics of the atom itself (we know how to solve that), and one that describes the internal dynamics of the atom, i.e. the electron orbitals. Solving the latter is a bit tricky, but not too bad. Once you arrive at a solution, you get some interesting properties that describe the hydrogen atom for free. So as you can see, we can actually describe quite a few systems analytically and the benefit of doing so is that we end up deriving properties that are intrinsic to the system, e.g. quantum numbers, energy levels (energy spectrum). Now there's so much more that I can go into, but that's the idea behind solving the Schrodinger equation. But why can't you do the same for the Helium atom? I mean, surely there must be some clever trick you can do! Well, the problem is that the Hamiltonian for the Helium atom is really a 3 body problem:
$$E\Psi({\bf{r_1,r_2}},t) = \left(-\cfrac{\hbar^2}{2m_1}\nabla_1^2-\cfrac{\hbar^2}{2m_2}\nabla_2^2-\cfrac{q^2}{4\pi\epsilon_0}\left(\cfrac{1}{r_1}+\cfrac{1}{r_2}\right)+\cfrac{q^2}{4\pi\epsilon_0|{\bf{r_1}-\bf{r_2}}|}\right)\Psi({\bf{r_1,r_2}},t)$$
The problem with such a system is that there aren't any tricks we can do to separate it into simpler constituents which we can solve individually. It's really the last term of the Hamiltonian that messes everything up. For the Hydrogen atom, we can separate the general solution into smaller parts, each of which describes some part of the system and is solvable. Ideally, we would like to get a solution that has one part describing the atom itself (free particle) and one part that describes the electron orbitals. This is, in fact, the case for all atoms with more than 1 proton-electron pair. In such cases, we result in approximations.
One of the simplest approaches would be to use something called $\textbf{variational methods}$. This is where you pick a Hamiltonian whose solution we know $\textbf{and}$ is similar to the Hamiltonian we are trying to solve, e.g. for the Helium atom, we would use the Hydrogen atom. The idea here is to first show that by doing this, we can bound the ground energy state (lowest energy) of the Hamiltonian we are approximating. This, in turn, leads to the following result:
$$E[\Psi] =<\Psi|\hat{H}_0+\hat{V}|\Psi> $$
where $\hat{H}_0$ gives us the lowest energy state for the Hydrogen atom. What this basically says is that we expect the energy of the Helium atom to be the ground energy of the Hydrogen atom, plus some extra contribution from the additional potential term. We then minimize the above functional, meaning we try to find the lowest value that it can take. And since this minimum value bounds the true value, we end up with a very accurate solution, in some cases the exact solution. Using the same principle, we can look for higher energy levels and eventually get the entire energy spectrum for the Helium atom. Another method we can use is called $\textbf{pertubation theory}$ which I suppose could be understood more intuitively. We'll stick to the Helium atom example. Suppose we can write the Hamiltonian for the Helium atom in the following way:
$$\hat{H}=\hat{H}_0 +\lambda\hat{V}$$
where $\lambda$ is a small parameter and $\hat{H}_0$ is a Hamiltonian whose solutions we know (i.e. the Hydrogen atom). The idea here is by doing some clever tricks, we seek a power series solution that converges to the real solution provided the necessary criteria are met. We end up with something along the lines of
$$E_n=E_n^{(0)}+\lambda V_{nm}+\lambda^2\sum_{k\neq n}\cfrac{|V_{kn}|^2}{E_n^{(0)}-E_{k}^{(0)}}+...$$
where $E_n^{(i)}$ are your $i$-th order correction terms. You pretty much try to add small corrections to the know Hamiltonian to get to the new one.
These are of course fairly simple examples and there are far more sophisticated methods out there. Both of the above have their own shortcomings and require special care when dealing with more complex problems. If the necessary conditions aren't met, then these methods will simply fail. Furthermore, in some instances, perturbation theory does not produce a convergent series, or if the series does converge, you cannot always guarantee a complete solution set. The great advantage to using such methods is that we have very powerful computers to do the job for us, but again that will depend on whether the problem can be feasibly solved in an ideal time period, i.e. for perturbation theory, you'd need to consider the radius of convergence to get an idea of how long the computation might take.
Now using methods like perturbation theory and variation is absolutely essential to make progress when analyzing complex Hamiltonians. In many cases, analytic solutions either don't exist or require some exotic mathematical tools which is a mountain all in itself. The theory of PDEs, especially non-linear PDEs which many Hamiltonian systems yield require theory from all branches of maths from elliptic functions to algebraic topology, and most of this theory is used to prove existence theorems rather than for computation.
On the other hand, analytic solutions give you the foundations you need to begin exploring. Although far more challenging, if successful they can give you far more depth and understanding of the system you're trying to study. You can find unexpected results lurking in the machinery which numerical approaches simply miss. Furthermore, it is essential to be $\textbf{rigorous}$ in your work, regardless if you do numerical analysis or take a more direct approach. Each step requires careful justification and when trying to understand complex systems, we shouldn't expect to always get simple results.
As for the other formulations of Quantum Mechanics, I'm not too sure what you're specifically referring to. If it's for example something like David-Bohm then the formulation seems to just add more complexity. I mean any formulation in most ways will be equivalent, since for it to be an acceptable theory, it needs to reproduce experimental observations. Modern Quantum mechanics has a very rigorous mathematical foundation, developed first by Hilbert and Von Neumann. Then further development was made by Dirac (his work is pretty much standard QM theory) and his work was similarly rigorously developed and fleshed out. We then have your standard QFT: fields, path integrals, renormalization, gauge theories, the standard model, etc. We also have algebraic QFT which is a very rigorous framework that studies Quantum Field Theories in curved space-time and is typically used when the underlying spacetime Manifold is curved. Of course, these frameworks deal with completely different things compared to QM.
