Contrary to the assumption in your question, understanding manifolds of dimension 5 and higher is considered to be easier than understanding 3- and 4-dimensional manifolds. Why? Because in $({\ge}5)$-manifolds there is more room to maneuver. Specifically, in a $({\ge}5)$-manifold any pair of embedded surfaces can be made disjoint by a small perturbation. This fact leads to the proof of the $h$-cobordism theorem and other tools for classifying $({\ge}5)$-manifolds.

I'm not aware of any convincing argument relating the above to the fact the space-time in 4-dimensional. But there is plenty of sub-convincing speculation on this topic.

Contrary to the assumption in your question, understanding manifolds of dimension 5 and higher is considered to be easier than understanding 3- and 4-dimensional manifolds. Why? Because in $({\ge}5)$-manifolds there is more room to maneuver. Specifically, in a $({\ge}5)$-manifold any pair of embedded surfaces can be made disjoint by a small perturbation. This fact leads to the proof of the $h$-cobordism theorem and other tools for classifying $({\ge}5)$-manifolds.

I'm not aware of any convincing argument relating the above to the fact the space-time is 4-dimensional. But there is plenty of sub-convincing speculation on this topic.