The Jewish calendar changed in 1513 BCE.
"It was a Tuesday."
From what I've seen on the subject, you're well away from evidence-based archaeology and deep into the realm of Biblical literalism.
It also marked a break with common regional tradition and a start of calculations based on national and cultural identity
There were no "calculations," or even any need for calculations, before the Babylonian Captivity and Diaspora. The Israelites were among several cultures that relied instead on terrestrial, ecological indicators for the start of spring (e.g. "seeing if a groundhog sees its shadow"). The Hebrew name of the Paschal month literally refers to the barley crop that was to be inspected (as per Exodus). The beginnings of months were reckoned empirically, as per current Islamic practice.
It was only after a non-negligible number of Jews lived too far away from the Temple (when there was a Temple) that the need for a computational calendar to maintain social cohesion (i.e. celebrating the same holidays on the same day) among the Diaspora presented itself. The modern Hebrew calendar relies on some decidedly Chaldean math that they likely picked up during the Captivity.
I know of no reason for Christians to keep track of any intercalary months, as Veadar (literally "and Adar [again]") ends by the time of any point of calculation for Easter (or more accurately, the memorial of Christ's sacrificial death)
If for no other reason than because Christians must be able to reckon the date of Easter months in advance in order to set the beginning of Lent, etc. Predicting the first full moon of spring (which defines the Paschal moon) requires some means of keeping track of lunations to know which new moon thirteen days preceding marks the proper start and to be able to count backwards the requisite numbers of days and weeks for the related movable feasts.
Predicting and setting the date of Easter requires knowing how many lunations pass between one Paschal moon and the next and how long each of them are. The Alexandrian computus settled upon by the early Church has always, necessarily, acted as a perpetual lunar almanac for the entire year (e.g. the Paschal moon is always the fourth of the year, and always 29 days long). The Gregorian method simply maintained as much of the tradition as Clavius saw feasible.
So they just call the next Sunday after that date "Easter Sunday"
The Ecumenical Councils determined that the theology of Easter demanded that it fall on a Sunday moreso than insisting that it be on Nissan 16, maintaining the symbolism of Jesus remaining dead through Saturday ("resting on the Sabbath") and rendering Sunday an "eighth day." Sunday is "the Lord's Day" (literally, in most European languages) specifically because it is the day of the week of the Resurrection.
It's not so much about precise intervals as it is about validating triggers.
On the contrary: for both the Jewish and Christian calendars it is more about the intervals than the triggers.
The defining astronomical events for the respective calendars and holiday schedules are necessarily instants (e.g. lunar opposition in Libra), and any given instant will fall on a different calendar date depending on the observer's longitude. The result would be, for example, Christians in Asia and the Americas observing Easter one week apart from each other, fragmenting the community.
There are two ways around this: declare a favored meridian ("lunar conjunction in Aquarius, Beijing standard time"), or to measure the intervening time with a standardized integer count of whole days. After all, lunar opposition doesn't actually occur the same amount of time after lunar conjunction, let alone exactly 1 123 200 s later. (Jews and Christians count mean lunations)