I prefer the following approach to understand why the frequency of the discrete-time sinusoid behaves so. As the confusion arises because of the continuous-time analogy lets repeat it here:
A continuous-time sinusoid $x(t) = \sin(2 \pi f_0 t) = \sin(\omega_0 t) $ is said to be periodic in $T_0 = \frac{1}{f_0} = \frac{2\pi}{\omega_0}$ where $f_0$ is frequency in Hz and $\omega_0$ is frequency in radians per second. It can be easily seen that since $f_0$, $\omega_0$ and $T_0$ are all continuous variables, there is no constraint on their interperetation: larger the $f_0$ (higher frequency) shorter will be $T_0$ (shorter period) for any $f_0$
But coming to the discrete-time case, a fundamental difference happens; the period $N$ of the discrete time sequences must be an integer and the smallest such integer period is $N_{min} = 1$ samples (in fact a period of $N=1$ samples corresponds to a DC signal and hence the minimim integer period for a sinusoid is effectively $N_{min} = 2$ samples.
Now what's the relation between the integer period and corresponding continuous frequency of the discrete-time sinuodid $x[n] = \sin(\omega_0 n)$ ? The answer comes from the relation $$x[n] = x[n+N]$$ which reveals
$$\sin(\omega_0 n) = \sin(\omega_0 (n+N)) = \sin(\omega_0 n + \omega_0 N) = \sin(\omega_0 n + 2\pi m) $$ resulting in:
$$ \omega_0 N = 2\pi m \longrightarrow \omega_0 = \frac{ 2 \pi m}{ N} $$
Putting $m=1$ for $N=2$ gives the highest frequency as $$\omega_{max} = \frac{ 2 \pi m}{N_{min}} = \frac{2 \pi}{2} = \pi $$
On the other hand the minimum frequency goes to zero as the maximum (positive) integer period goes to infinity $N_{max} \rightarrow \infty$ hence
$$ \omega_{min} = \frac{ 2\pi m}{N_{max}} \rightarrow 0 $$ for any finite $m$.
This way it can bee seen that discrete-time frequencies (which are fundamentally related to allowed range of discrete-time periods $N$) begin from $\omega=0$ as the minimum frequency and reach up to maximum frequency of $\omega = \pi$ for the minimum allowed integer period of $N_{min}=2$ samples.