First have look at my previous answer
What is the Wilsonian definition of renormalizability?
for general explanations about UV renormalization and what one is trying to achieve. The new question relates to a very simple phenomenon in the the theory of ODEs and PDEs, in regards to global-in-time well-posedness. Consider a differential equation
$$
\frac{dx}{dt}=\epsilon x^2
$$
where $\epsilon=\pm 1$.
In the focusing case $\epsilon=1$, the solution goes to infinity in finite time. In the defocusing case $\epsilon=-1$ one has the opposite phenomenon of "coming down from infinity". The solution becomes infinite if we run the equation backward in time. A sign of that is for $t_1<t_2$, one has a uniform bound on $x(t_2)$ regardless of the size of the initial condition $x(t_1)$ set at time $t_1$.
Here $t=-\log\Lambda$ where $\Lambda$ is the UV cutoff. The problem of UV renormalization is not about solving the ODE (given by the beta function) forward in time, but rather deciding on the value we want say for $x(0)$, an effective coupling at say unit scale, and then figuring out the initial condition $x(t)$ at $t=-\infty$ which allows us to realize the wanted value a time zero. For $\phi_4^4$ we are in the coming down from infinity situation, where for any $x(0)>0$ we find that there is no suitable initial condition at $t=-\infty$ because the solution diverges before, at a finite negative time. The only way to reach $t=-\infty$ is to take $x(0)=0$, i.e., when the theory obtained at the end of the day (after removing cutoffs) has zero effective coupling, i.e., is trivial. Note that all this is due to the sign of $\epsilon$, i.e., the bubble graph $>O<$. It has the opposite sign for gauge theories and Fermionic models like Gross-Neveu, hence asymptotic freedom and escaping this mechanism for triviality. Note that $x(0)<0$ would also work if one can define $\phi_4^4$ for negative coupling. This was done nonperturbatively, but only for planar graphs, by 't Hooft and Rivasseau.
