It is easy to show that $\mathbb{N}$ with the cofinite topology is not path connected and that any set with cardinality $\geq 2^{\aleph_0}$ equipped with the cofinite topology is in fact path connected.

what about cardinalities $\aleph_0<\alpha<2^{\aleph_0}$ (under the assumption that such exist obviously)?

If $\alpha $ is path connected then any cardinality $\geq \alpha$ is also, so an interesting direction would be trying and checking whether $\aleph_1$ is path connected (I have no clue about how one can even start checking this).

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