Consider $(S^1 \times \Sigma^2, g)$, where $g$ is any Riemannian metric on the compact and closed $3$-manifold $S^1 \times \Sigma^2$.
Question: Does there always exist a nowhere vanishing harmonic $1$-form on $S^1 \times \Sigma^2$? If the answer to this question is No, how about the generalisation to $k$-parameter families of metrics?
So far I tried to find an example of a harmonic $1$-form on $T^3=S^1 \times S^1 \times S^1$ that does have a zero but did not succeed.
I have cross-posted this question to: https://mathoverflow.net/questions/407340/nowhere-vanishing-harmonic-1-forms-on-3-manifolds.
This is not a full answer, but I just wanted to add holonomy to what you already found and it became to long for a comment.
The Bochner Theorems say, that for a Riemann manifold $(M,g)$ with $Ric \geq 0$ any harmonic $1$-form is parallel.
In this case, if $\omega$ is harmonic and there exists $p \in M$ with $\omega_p = 0$ then $\omega = 0$ globally. That is why you cannot find non-trivial harmonic $1$-forms on $T^3$ having zeroes.
If $Ric = 0$, then $1$-forms are harmonic iff they are parallel. Then finding a nowhere vanishing harmonic $1$-form is the same as finding a nontrivial parallel vector field, which exists if and only if $$ Hol((M,g)) \subset SO(m-1) \subset SO(m).$$
So the metrics constructed in Calabi's Theorem you linked for nontrivial transitive $1$-forms with zeroes, cannot have $Ric \geq 0$ globally.