Derivation of the lowpass to bandpass transformation
You're right that the multiplication of a low pass and a high pass filter results in a band pass filter, as long as the cut-off frequency of the low pass is higher than the cut-off frequency of the high pass. The problem with that approach is that low pass and high pass filters with magnitude responses that are optimal according to some chosen criterion (Butterworth, Chebyshev, Cauer) will not result in an optimum band pass filter.
On the other hand, mapping a single optimal filter will result in another optimal filter. Using $\omega_l\omega_u=\omega_0^2$, where $\omega_l$ and $\omega_u$ are lower and upper band edges, respectively, and $\omega_0$ is the center frequency of the band pass filter, and leaving out constants for the sake of simplicity, the transformation can be written as
$$s\longmapsto \frac{s^2+\omega_0^2}{s}\tag{1}$$
[Note that $\omega_l$ and $\omega_u$ are denoted as $\omega_1$ and $\omega_2$ in the OP, but they're used in a different way in the figure below.]
The mapping $(1)$ maps DC ($\omega=0$) to the desired center frequency $\omega_0$. Furthermore, $s=\pm\infty$ is mapped to $s=0$ and $s=\infty$. So the whole frequency axis of the low pass filter is mapped to the positive frequency axis of the band pass filter. (The same is true for the negative half axis of the band pass filter):

(from: Digital Filter Design by Parks and Burrus)