Jesus and His disciples didn’t really need to pay the half-shekel Temple Tax. Jesus and Peter paid it, however, so as to not offend the Jews’ religious sensibilities (see Mt. 17.24-27). Jesus, of course, didn’t need “to make atonement for His life” (Ex. 30.16c ESV). Also, by logic (since it is God’s Temple), His redeemed children are free from paying it.

The yearly half-Shekel, as I understand the text and motive, was an enrollment in Israel, presumably, for the time of Yom Kippur at the Temple where the yearly atonement sacrifice was given (see Lev. 16). Ex. 30.11-16 establishes the practice of paying the half-Shekel whose ultimate monetary purpose was the upkeep of the Jewish Temple (Tabernacle in this case). This upkeep was vital since, at least liturgically, the Israelites needed to be represented and remembered by God: that it may bring the people of Israel to remembrance before the LORD, so as to make atonement for your lives (vs. 16 ESV). Since this instruction was for the purpose of atonement, the collection must have occurred before Yom Kippur. Heb. 10.1-14 notes that the Mosaic Law (in this case the sacrifices instead of human regulations), required a yearly sacrifice that reminded the people of last year’s sins (vs. 3). While this section speaks of the sacrificial system in general, vs. 3 specifically has to be the Yom Kippur observance: But in these sacrifices there is a reminder of sins every year (ESV).

Heb. 10.7 quotes the Messianic Psalm 40, and notes that Jesus’ willing obedience cut the second (New) covenant: then he added, “Behold, I have come to do your will.” He does away with the first in order to establish the second [covenant] (Heb.10.9 ESV). Heb. 10.12 is a key verse that explains implicitly why Christ’s one-time sacrifice was sufficient for sin: But when Christ had offered for all time a single sacrifice for sins, he sat down at the right hand of God (ESV). Since Jesus is seated now in heaven in His resurrected human body, this means He will judge from that throne. The Father has accepted the Son’s sacrifice for His people (see Jn. 16.10, Acts 17.31) is the reason Jesus is now seated on His Father’s throne.

The author of Hebrews contrasts the Mosaic Law’s symbolic nature with the eternal nature of King Jesus’ act for His children: And every priest stands daily at his service, offering repeatedly the same sacrifices, which can never take away sins. But when Christ had offered for all time a single sacrifice for sins, he sat down at the right hand of God, waiting from that time until his enemies should be made a footstool for his feet. For by a single offering he has perfected for all time those who are being sanctified (Heb. 10.11-14).