Is QFT actually relativistic?

In any interactive quantum field theory in four dimensions, interactions are modeled as products of distributions at single space-time points.

To my understanding, there is no sense in which QFT can actually be made relativistic in the first place:

• The expression $\lambda\hat{\phi}^{4}(x)$ of any product of two fields at the same point is completely meaningless and does not exist.
• It is not possible to define the theory perturbatively, as because of Haag's Theorem there is no unitary operator $\hat{U}_{i}$ similar to how it is defined in perturbative quantum mechanics. Even after renormalizing the various perturbative terms, the full sum cannot converge to any value whatsoever, so it's not even clear in which sense this is a perturbative approximation anyway.
• If Haag's Theorem is evaded by working on a discrete space-time lattice, the theory is clearly not covariant anymore, and the continuum limit cannot possibly exist because you would obtain infinitely many degrees of freedom and the theorem would again apply.

If the lagrangian for an interactive theory cannot even be written down coherently, the perturbative approach is not mathematically defined, and the lattice theory is not covariant, in which sense is QFT even a relativistic theory of quantum mechanics?