The diffractive electroproduction of heavy quarkonia (V) is an effective tool to study the structure of $V\rightarrow Q\bar Q$ transition. The most of existing studies in the literature are based on the unjustified assumption of a similar structure of both $\gamma\to Q\bar Q$ and $V\to Q\bar Q$ vertices, typically performed in the light-front frame. Such the photon-like $V\to Q\bar Q$ vertex, besides the $S$-wave component, also contains an extra $D$-wave admixture in the $Q\bar Q$ rest frame. However, the relative weight of this contribution cannot be justified by any reasonable nonrelativistic $Q-\bar Q$ potential model. Consequently, the recent model predictions for heavy quarkonium photoproduction cross sections are thus contaminated by these $D$-wave effects, which may lead to serious problems with a correct interpretation of the experimental data. In this work, we investigate and discuss the relative role of the $D$-wave contribution by comparison of our predictions based on the photon-like structure with results within a simple $S$-wave-only form of the quarkonium vertex. Calculations performed in the color dipole formalism are tested by available data. We have found that the production of radially excited heavy quarkonium states is more effective for the study of the $V\to Q\bar Q$ vertex structure due to a stronger sensitivity of the undesirable $D$-wave contribution to a nodal structure of quarkonium wave functions.