We do not know if spacetime is quantized or not, and there are theories claiming that it is (LQG etc). And sure, we don't have a coherent "theory of everything" that includes quantized spacetime, and the models that we do have are contradictory if spacetime is quantized. But we already know that those models are deficient.
Our current theories only work if spacetime is not quantized. Any theory that tries to quantize spacetime needs to somehow replace special relativity and its Lorrentz transform with something completely different, while still remaining consistent with the huge number of observations that confirm SR works - especially the extremely precise experiments that confirmed QFT. This is one of the reasons why LQG is almost certainly wrong, by the way.
Note that this is separate from the problem with GR-QFT inconsistencies. All of our current theories are based on and only work if spacetime is continuous. While it's not impossible that a new theory with quantized spacetime could exist and work, it's not at all required.
The one thing about spacetime that we do believe might be quantizable, and would have to be quantized for GR and QFT to be compatible, is the curvature of spacetime. But even if spacetime can only be curved in discrete quanta, that would not mean that position would be quantized.
