I am not really sure how to answer this, because as far as I know infant sacrifice via live burial under houses was never practiced in the ancient world. The practice of infant sacrifice is certainly known, with perhaps Carthage being the most infamous example, but archaeological and literary evidence implies that this was sacrifice by cremation, not burial under homes. Even still, there is disagreement over whether the cremated remains represented sacrificed infants, or natural deaths and subsequent cremation. The famous “bog bodies” of Northern Europe are another good example of human sacrifice, but only around 10% of these victims were children. These individuals were typically killed by garrotting, strangulation, and/or other violent attacks such as stabbing before being lowered into a bog or other body of water.

Infanticide was far more common than infant sacrifice, and a 1976 survey found that it was practiced in 80% of 393 societies which were included in the study, although the actual number may be higher as evidence of infanticide is not always easy to find. The important difference between these two practices is that sacrifice was a religious ritual in which an infant was offered in exchange for something, whereas infanticide was the unceremonious disposal of unwanted children. Both are gruesome, even chilling, but the same Greeks and Romans who would readily abandon or kill an unwanted infant would be mortified at the idea of making a human sacrifice. It is important to note that in many ancient societies, infants were not accorded full personhood until they reached a certain age, and the abandonment (often referred to as “exposure”) of infants was much more palatable.

There were many factors which motivated infanticide in the ancient world. Families might engage in infanticide as a means of family planning, eliminating extra mouths to feed. There were also social pressures which prompted the killing or exposure of illegitimate, malformed, disabled, or otherwise stigmatised children. Gender preferences influenced the decisions of parents in many cases, where preference for a boy or girl meant the selective care of certain offspring. These factors often overlapped, as disabled children were less capable of supporting their parents in old age and may have required additional care.

The widespread evidence of infanticide from the ancient world is striking. For example, archaeological evidence from Roman era Britain indicated the regular practice of infanticide, especially when contrasted with the more natural ratios of Medieval infant burials. The Roman era burials were disproportionately made up of full term fetuses, rather than the wide age range that would be expected from fetal and infant mortality. Similar trends were noted at several other pre-Roman and Roman sites in Europe, and attest to the frequency of this practice.

Urban infanticide was certainly practiced, and there is archaeological evidence of infants being disposed of in urban settings. For example, in Roman Ashkelon, modern day Israel, the remains of nearly 100 infants were found to have been disposed in a sewer drain near a bathhouse, presumably the unwanted children of prostitutes working in a nearby brothel. Notably, these infants were disproportionately male, implying that girls were kept to be trained into the profession of their mothers.

Although there were social pressures which made female offspring less desirable to Greek and Roman parents, there is disagreement as to whether there were ever high rates of female infanticide. Economic reasons are usually cited as an influencing factor Daughters could not usually earn large sums, could not inherit property and in order to marry them off, parents had to pay a dowry, making them a potential cost-sink. At the same time, girls could help support the household through crafts such as weaving, domestic labour, and even perhaps work outside of the home. It has also been pointed out that too many male children would require the family’s property and inheritance to be divided too many times, providing an economic incentive for male infanticide.

Infanticide played an important role in Greek and Roman myth, from Cronus’ cannibalisation of his children, to Hera's attempted murder of Dionysios, to Telthusa’s decision to pass her daughter Iphis off as a boy so that her husband would not expose her. Greek and Roman literature (including plays, poetry, and histories) also attests to the real world practice of infant exposure, but this was not the case everywhere. Greek and Roman authors claimed that Egyptians did not practice infanticide, and archaeological evidence bears this out. In fact, it seems that most Greek settlers who immigrated to Egypt during the Ptolemaic period abandoned this custom within a few generations, although papyrological evidence does attest to the legal and social echoes of this tradition.

Beyond this, infant mortality claimed 30-40% of infants in the ancient Mediterranean, meaning that families were naturally limited. This makes it hard to create stable estimates of the prevalence of infanticide throughout time.

Finally, ritual killing by burial was practiced in the ancient world, but not too commonly. One example off of the top of my head is the Roman punishment for Vestal Virgins who broke their vows. The priestesses of Vesta had numerous responsibilities and restrictions, and if one was found to have broken her vows she was buried alive in order to appease the goddess Vesta. Another example are instances of wartime human sacrifice from the Roman Republic. These sacrifices involved the live burial of couples of Gauls and Greeks, although Zsussannah Varlhelyi noted parallel burials of non-Roman enemies or symbolic enemies prior to the Late Roman Republic.

But these are anomalous, highly unusual occurrences in a society which viewed human sacrifice with contempt, disgust, and outright horror. In Rome, legislature was passed which outlawed human immolatio in 97 BCE, although this was bloody sacrifice not specifically burial, there is no evidence that live burial was in any way common. And of course, those examples I mentioned above involved adults not infants.

It has been suggested that the Classical literary evidence historians have to work with does not reflect earlier attitudes, and there is some evidence that human ritual killings was practiced to some extent in many ancient societies around the Mediterranean and Near East. Having said that, I really do not have the time to dig into it right now unfortunately. Suffice to say, that the specific scenario you are describing was far from common, in all likelihood non-existent.

There is a lot more I could write on this, but I am up to my neck in irl work, and hopefully between this answer and the sources below you will find the information you are looking for.

#Sources:

Tracing Childhood: Bioarchaeological investigations of early lives in Antiquity edited by Jennifer L. Thompson, Marta P. Alfonso-Durrity, and John J. Crandall

Constructions of Childhood in Ancient Greece and Italy by Ada Cohen and Jeremy B. Rutter

The Problem of Female Infanticide in the Greco-Roman World by Donald Engels

Not Worth the Rearing: The Causes of Infant Exposure in Ancient Greece by Cynthia Patterson

The Specters of Roman Imperialism: The Live Burials of Gauls and Greeks at Rome by Zsuzsanna Varlhelyi