I somehow "proved" that given any LTI system, its transfer function has to be constant. What am I missing?
When the input to an LTI system is an eigenfunction, then:

$$ x(t) = e^{j\omega_0 t} =&gt; y(t) = H(\omega_0) e^{j\omega_0 t}$$

where $H(\omega_0)$ is the associated eigenvalue of the LTI system; the transfer function evaluated at the particular frequency:

$$ H(\omega_0) = H(s)|_{s = j\omega_0}. $$

Here not the transfer function $H(s)$, but just one value of it is found by just one eigenfunction.

But if you use the general expression for the eigenfunction, then the input is $x(t) = e^{j\omega t}$ and the output will be $$y(t) = H(j\omega) e^{j \omega t}. $$ Here you assume that $H(j\omega) = a$ ; a constant,  and therefore in the fraction expression you will find out that $H(s) = \frac{Y(s)}{X(s)} = a = H(j\omega)$. Here, the constant $a$ is actually the unknown variable $H(j\omega)$...

Note that the eigenvalue $H(\omega)$ will be a complex-constant for a given particular frequency $\omega_0$. But as the given frequency changes, the constant (in general) will change too. It's just like an ordinary function $f(x)$ treated as a single value for a particular $x = x_0$ but a rather a varying mapping for general $x$ values...