In a renormalizable theory (with or without Landau pole), the perturbative results change only by $O(\Lambda^{-1})$ when you move the cutoff to $\infty$ and adjust the finitely many counterterms accordingly. Thus one could always work at some finite $\Lambda$ and stay within the experimental bounds.

However, Poincare invariance (and in not gauge invariant renormalization schemes also gauge invariance) is then violated by $O(\Lambda^{-1})$, too. This means that $\Lambda$ has to be large enough to respect the very stringent experimental bounds on a violation of Poincare invariance.

Thus the limit $\Lambda\to\infty$ is primarily needed to have exact symmetries. In a sense, this means that it is needed only for theoretical reasons of elegance, since if Poincare invariance were broken it would call for an explanation why it is so extremely well satisfied.

Nonrenormalizable theories show precisely the same behavior, except that infinitely many counterterms are needed to get a finite limit as $\Lambda\to\infty$. This is not a disaster for predictability as all but a few of these counterterms are suppressed by high powers of the mass scale of the theory (such as the Planck mass for gravity).

All this is valid on the perturbative level.

Nonperturbatively things could go wrong if there is a Landau pole, but the existence of a Landau pole for $\Phi^4$ theory or QED is proved only in low order perturbation theory. Hence in fact nothing mathematically convincing is known about the obstructions to nonperturbative QFTs (whose observable fields satisfy the Wightman axioms).

My personal belief is that QED (or at least a variant of QED that contains nuclei in addition to electrons, with simplified interactions and conserved nucleon number - i.e., excluding radioactivity) must exist since it is the most accurate theory that we have.