Control design: under what conditions can closed-loop poles be placed arbitrarily?
Unnecessary additional information that might help

From a pragmatic point of view, there might be a problem when trying to control a given plant using only its transfer function $G_p(s)$. In general, you test the plant putting a series of inputs and seeing how it reacts, measuring the respective outputs. From that, you can model the plant as a transfer function in the Laplace domain. However, the intrinsic dynamics of the plant may have hidden unstable modes (for instance, poles that cancel with zeros, which means you are not able to notice their presence in the system) and thus the control system you design wouldn't work as expected.

But that's in real life. In Control Theory, we are often offered the transfer function of the plant with all the poles and all the zeros. So let's assume that $G_p(s)$ doesn't show this weird behaviour I described above. In that case, one must notice (at least) three advantages that state spaces (SS from now on) offer:


Working with SS instead of transfer functions can be way easier. If the order of the system is too high, operating with matrices can be the only way to deal with the system.
SS provide an easy way to deal with MIMO systems. Using transfer functions for each pair of input-output would be too tedious.
SS let us see those hidden unstable modes I referred to before (looking at the eigenvalues of the matrix $A$), while in the transfer function they cancel out.


The actual answer to your question

Now getting to your question, you've asked what happens when the system is of low order (in your example, the plant is of second order) and SISO. 

You can definitely find your transfer function performing the variable isolation you've written for $G_c(s)$. The problem with that is that some things could go wrong. For example,


$G_c(s)$ could be impossible to implement.
$G_c(s)$ could be unstable, thus making the control system non-sense.
$G_c(s)$ could have zeros in the RHP that might cancel out with poles of $G_p(s)$.


If none of those things happen (maybe there are some other aspects that one should take into account, but I couldn't think of any additional one), then go ahead: you are free to find your control system, $G_c(s)$, without using full-state feedback at all.

The thing is that SS offer us a lot more information than transfer functions do. This is easy to notice, as when working with SS we are looking at not only the input and output variables, but the other state variables as well, which are totally left apart when working with transfer functions and block diagrams alone. When doing full-state feedback, we are using this additional information to control the system (i.e. you don't just use the output, but each one of the state variables). This means that there are other tools we can use to stabilize the plant and we were not using before. If the design using full-state feedback is good, then the closed-loop will work as a charm and none of the problems I've described above (unstability, unstable poles and zeros cancelling out, etc.) will appear.