I will start by posing yet another question, given a system \$H(s)\$ why can't we determine a compensator \$C(s)\$ such that
$$ C(s)H(s) = P(s)$$
where \$P(s)\$ is the desired system? Well, although it is possible to mathematically find some \$C(s)\$ to do that, it can be undesirable to do so. As \$C(s) = P(s)H^{-1}(s) \$ can be unstable, and if we calculate \$C(s) = P(s)\hat{H}^{-1}(s) \$ based on an estimate of the system, \$\hat{H}(s)\$, which differs slightly from \$H(s)\$, will also lead to problematic performance (and maybe even instability too.)
Now, going back to the question of finding an arbitrary \$C(s)\$ to have
\$ \frac{C(s)H(s)}{1+C(s)H(s)} = P(s)\$, we have similar problems that won't be clear by looking at \$\frac{C(s)H(s)}{1+C(s)H(s)}\$ but will be perceived by looking at the Bode plot, Nyquist plot and so on of the resulting closed-loop system, as you might find that the "desired" \$C(s)\$ that makes \$ \frac{C(s)H(s)}{1+C(s)H(s)} = P(s)\$ will only work if you know \$H(s)\$ perfectly, and for any deviation from \$H(s)\$ to the real system will lead to a narrow, or no, gain/phase margin in the experimental Bode plot and others.
The issues will likely show up on the "gang of six" (see Aström & Murray Ch. 10 onward, but in particular section 12.1. Sensitivity Functions, but also 9.4 on Pole/Zero Cancellations) which are sensitivity functions and indicate how likely your current closed-loop configuration is to be affected by noise being injected in different parts of the loop, i.e., you can have sensor noise altering the measured output or have noise in the actuator changing the system in a way different than what the controller actually aimed at. The "gang of six" are the transfer functions modeling the impact of \$(r,v,w)\$ on \$(u,y)\$, where \$r\$ is the reference of the control system, \$v\$ is the noise injected on the control effort \$u\$, and \$w\$ is the noise injected on the output \$y\$.
An example, given the system \$ H(s) = \frac{1}{s-10} \$ we desire that it behaves as \$P(s) = \frac{1}{s+100}\$ after adding the feedback control. So we use your idea and obtain that the controller should be \$C(s) = \frac{s -10}{s+99}\$. The controller is a stable transfer function, and
$$ \frac{C(s)H(s)}{1+C(s)H(s)} = \frac{\frac{s -10}{s+99}\frac{1}{s-10}}{1+\frac{s -10}{s+99}\frac{1}{s-10}} = \frac{\frac{1}{s+99}}{1+\frac{1}{s+99}}= \frac{1}{s+100}=P(s)$$
also checks out, and will be stable. But now lets look at the load disturbance to measurement signal, that is, how a noise in the actuation \$u\$ affects the output \$y\$. The transfer function will be
$$ G_{y\xleftarrow{}u}(s) = \frac{H(s)}{1+C(s)H(s)} = \frac{\frac{1}{s-10}}{1+\frac{s -10}{s+99}\frac{1}{s-10}} = \frac{\frac{1}{s-10}}{1+\frac{1}{s+99}} = \frac{s+99}{(s-10)(s+100)}$$
which is unstable. So any small error in the actuation, say noise in your amplifier or motor, will cause the system to behave badly.
