The answer should be "no" for the 0-1 law. Take, for example, this sentence:

"What is the proportion of square-free natural numbers?"

The answer is known to be $6/\pi^2$

Then turn it into a question about finite cyclic groups:

"How many cyclic groups $G$ of order less than $N$ have the property that none of its non-unit elements $g\in G$ is a square root of unity, $g\ne e, g^2=e$?"

Finally, define property $\theta$ for a finite group $G$ as simultaneously being cyclic and having no square roots of unity.

The answer should be "no" for the 0-1 law. Take, for example, this sentence:

"What is the proportion of square-free natural numbers?"

The answer is known to be $6/\pi^2$

Then turn it into a question about finite cyclic groups:

"How many cyclic groups $G$ of order less than $N$ have the property that none of its non-unit elements $g\in G$ is a square root of unity, $g\ne e, g^2=e$?"

Finally, define property $\theta$ for a finite group $G$ as simultaneously being cyclic and having no square roots of unity.

EDIT: as pointed out in the comments, $p(\theta)=0$ because the denominator remains "all groups of order $\le N$", not "cyclic groups of order $\le N$".