I totally understand where you got this idea and it is very logical. To deal with the first part of your question and the relation of the cult of saints to the deification of emperors, I would be hesitant to draw too much of a link. The exact theological nature of the divine emperors is complicated, but they seem to have been treated in a somewhat similar manner to other culture heroes like Hercules who, through their great deeds, had gained a sufficient level of renown to ascend, become godlike and worthy of human worship. Saints meanwhile, were honoured specifically because they were dead. While their souls ascended to heaven, their bodies remained firmly placed on earth and the soul remained attached to it. The spirit of the saint was in a metaphysical sense present in their tomb. The holiness of heaven then flowed out of the body of the saint, blessing all those around them. This was a fact which was noted and, it must be said, rather weirded out, their pagan opponents. Eunapius of Sardis, for instance, commented rather acidly in the 4th century AD that Christians “collected the bones and skulls of criminals who had been put to death for numerous crimes… made them out to be gods and though that they became better by defiling themselves at their graves.” Their physical bodies then provided that link up to heaven whereby people could gain access to the divine.

Now, as for the thrust of your question about the cult of saints being a continuation of Roman religious practice. This is complicated. On the one hand, Christians did not emerge from a vacuum, but rather from the Roman society which surrounded them and they brought with them all the cultural baggage of that same society. This included many rituals surrounding the dead, worship of the dead at a family shrine and beliefs that the saint was like a Daimon or guardian spirit. I’m afraid that I don’t know too much about these practices so I will let someone else with greater knowledge than myself fill in the details here.

On the other, hand, Christians operated in a very different cultural landscape to their pagan neighbours and this should make us cautious of drawing too many analogies with previous religious practice. A few things to note. The veneration of ancestors in the Roman world was very much a family thing. The tomb of the saint, meanwhile, was public. This might be partially explained historically by the fact that Christianity constituted an artificial kin group – joining the Jesus cult in some ways severed the ties of family and community which a person was embedded in prior to their conversion. Further, Christians quite decisively broke from Roman practice by breaking the separation between the cemetery and the rest of the community, integrating the two in a way which their Roman neighbours found strange. Who, after all, would like to live in a graveyard?

The final thing to add here is that historians such as Peter Brown have argued that it was the Cult of Saints which made Catholicism and not the other way around. The shrine of a local martyr or holy man, with the accompanying processions and so on, were already the focus of many late antique Christian communities. It appears that it was a deliberate strategy by many enterprising bishops to associate themselves with the shrine and control access to it – and thus to the divine. This in turn increased their own power and prestige. Not to put too fine a point on it, the official church hierarchy appropriated the belief surrounding saints as part of a power grab by an emerging church hierarchy.