First off, sorry for the late response to this question. I saw it a little while back and one thing led to another and pushed it out of my mind. I'll preface this by saying that I am more familiar with the early medieval manifestations of various saint cults, but I believe I am familiar enough with developments in Late Antiquity to be able to answer your question.

The rise of the cult of the saints is -for me at least- probably the most fascinating development in late ancient Christianities, if for nothing else than its almost ubiquitous popularity across Late Ancient/Early Medieval society and the downright wild stories that come out of the genre of saints' lives (hagiography) If you are interested in further reading, the foundational work on the subject is Peter Brown's The Cult of the Saints: Its Rise and Function in Latin Christianity; it's a great place to start.

To your question: no, I do not think that we can understand the rise of veneration as a direct continuation of the role played by gods and heroes in pre-Christian Mediterranean cults. The criticism of saints as one for one replacements of pre-Christian deities has its roots in Protestant refutations of the practice, as well as in Enlightenment writings aimed at thumbing their noses at the medieval Church (looking at you Thomas Paine). It doesn't help that late ancient pagan contemporaries also thought it funny -or rather horrifying- that Christians were treating their dead as gods.

I would hazard that the reason for the confusion is the large degree of cultural and religious syncretism that does take place. But to some degree this is to be expected. The integration of various rites of veneration, symbolism, or even literary tropes into new practices or institutions is fairly common when describing cultural and religious change, it does not mean that the institution itself is virtually the same. While there are certain similarities between the cult of the heroes and gods of antiquity and the cult of the saints, the analogy breaks down fairly quickly when we consider the actual role played by the saints in the Christian cosmology.

An important development in the Christianization of the Mediterranean is the redrawing of the subtle boundaries between the living, the dead, and the divine. 'Pagan' understandings often maintained a stark divide between the divine and the dead; the taint of death barred access to holy space. By contrast, the importance of death in Christian cosmology, manifested in the 'good deaths' of the martyrs, actually opened a channel by which death and holy suffering qualified the individual to have special access to the divine. The martyrs were venerated because of their proximity to God, brought about by their suffering and special deaths. The focus on death and the bodies of saints was actually a point of derision and disgust for the same pagan writers who likened the practice to their own forms of worship. Julian the Apostate remarked:

The carrying of the corpses of the dead through a great assembly of people, in the midst of dense crowds, staining the eyesight of all with ill-omened sights of the dead. What day so touched with death could be lucky? How, after being present at such ceremonies, could anyone approach the gods and their temples?

Late ancient pagan understandings of the cult of the saints were framed as a perversion of religious practice, a disgusting act of defilement through close association with the dead. Precisely because saints were human (which despite their elevation, a point that is never lost) and suffered a mortal death, they were able to intercede between everyday Christians and God. The cult of the heroes, in which humans became quasi-divine ,lacks the focus on death and suffering. In short, the emphasis on the humanity and suffering of the saints provides a distinct break from earlier veneration of heroes and gods, from which the stain of death and mortality is crucially absent.

Again, the role played by saints in late ancient Christian society does not preclude influence from earlier religious practices, nor does it bar imagery and other pre-Christian elements of the divine from making their way into the stories and legends surrounding saints. St. Brigid in Ireland certainly shares some characteristics with a pre-Christian goddess of the same name (though the Irish were always kind of odd). However, cultural syncretism does not imply a one to one relationship or any sort of direct correlation, and I am not aware of any direct relationship between a Greco-Roman god and a saint; their role and association with death rendered the saints fundamentally distinct from their pagan colleagues.