The first reason is that you do not gain much by considering major arcs around rational numbers with denominator $\geq 3$. The reason is that the contribution of the major arcs $\{a/q:(a,q)=1\}$ reflects the inhomogenity of the distribution of the function in question modulo $q$, which cannot be explained by the distribution modulo proper divisors of $q$. But already $q\geq 3$ this inhomogenity is pretty small. Define $r_q(a)$ as the number of representations of $a\pmod{q}$ as the sum of 5 residue classes coprime to $q$. Then we have $r_3(1)=r_3(2)=11$, $r_3(0)=10$, and the difference of these values is pretty small. As $q$ increases, this difference decreases pretty fast.

The second reason is that a good numerical evaluation of an arc around $a/q$ requires bounds for the error term in the prime number theorem modulo $q$, and for small values (say around $10^{30}$) such estimates are surprisingly difficult.
Note that for Helfgott's proof of ternary Goldbach computations for the roots of Dirichlet $L$-series were necessary, which went way beyond everything done before in this direction.