The inequalities you have included in your post are the so-called MTZ SEC's which are known to be very weak. That is, they do not provide a very tight approximation of the convex hull of integer feasible solutions. There are tons of papers providing polynomial-sized sets of SEC's (see for instance Bektaş and Gouveia (2014)1 for stronger versions of the MTZ or Letchford and Salazar-González (2019)2 for (multi-)commodity flow-based formulations of the SEC's).
However, in practice, for problems consisting of more than say 20-30 customers, it usually works better to separate stronger SEC's from families of inequalities of exponential sizes. For example, for the CVRP, there is the set of "rounded capacity inequalities" given by
\[
\sum_{i\in S}\sum_{j\not\in S}z_{ij} \geq \left\lceil \frac{\sum_{i\in S}q_i}{Q}\right\rceil,\qquad \forall S\subseteq \{1,\dots,n\}
\]
where $q_i$ is the demand of customer $i$ and $Q$ is the capacity of the vehicles (for information on the strength of different types of SEC's you can take a look at the very nice paper by Letchford and Salazar-González (2006))3.
I believe that the state-of-the-art solvers for vehicle routing problems combine the agility of strong cutting planes with the strength of the LP bound obtained from the relaxation of a set-partitioning based formulation of the VRP in question.
References
[1] Bektaş, T., Gouveia, L. (2014). Requiem for the Miller-Tucker-Zemlin subtour elimination constraints? European Journal of Operational Research. 236(3):820-832.
[2] Letchford, A. N., Salazar-González, J-J. (2019). The Capacitated Vehicle Routing Problem: Stronger bounds in pseudo-polynomial time. European Journal of Operational Research. 272(1):24-31.
[3] Letchford, A. N., Salazar-González, J-J. (2006). Projection results for vehicle routing. Mathematical Programming. 105(2-3):251-274.