A common Evangelical answer to your question would be that even though Christ's work on the cross was* to redeem mankind from the effects of the fall, there is a now-and-not-yet-ness in regard to the complete outworking of the victory won at Calvary (*ie 'was' is really 'was'/'is'/'will be') . In particular, the physical creation in general and our mortal bodies in particular will be amongst the last to experience full redemption from the effects of the fall, and will do so only upon Christ's return in glory:
Present Suffering and Future Glory
18 I consider that our present sufferings are not worth comparing with the glory that will be revealed in us. 19 For the creation waits in eager expectation for the children of God to be revealed. 20 For the creation was subjected to frustration, not by its own choice, but by the will of the one who subjected it, in hope 21 that the creation itself will be liberated from its bondage to decay and brought into the freedom and glory of the children of God.
22 We know that the whole creation has been groaning as in the pains of childbirth right up to the present time. 23 Not only so, but we ourselves, who have the firstfruits of the Spirit, groan inwardly as we wait eagerly for our adoption to sonship, the redemption of our bodies. 24 For in this hope we were saved. But hope that is seen is no hope at all. Who hopes for what they already have? 25 But if we hope for what we do not yet have, we wait for it patiently. - Romans 8:18-25 NIV
Paul seems to be saying here that because mankind fell and suffered the consequences of sin as a curse, the rest of creation (which had been committed to the stewardship of mankind) was subjected to 'frustration' in order that it's 'groans' could testify to the need for the redemption and glorification of it's fallen race of stewards - this is a theological rationale for the current imperfections in nature.
Compare also 1 Corinthians 15:51-55 which speaks of the Resurrection to come as an overcoming not just of death, but also of the 'perishable' or 'corruptible' nature of our bodies - new resurrected bodies will be imperishable and incorruptible; and further, in the New Jerusalem:
‘He will wipe every tear from their eyes. There will be no more death’ or mourning or crying or pain, for the old order of things has passed away. - Revelation 21:4 NIV
So a complete deliverance from pain awaits the passing away of the old order.
An alternative (currently minority) perspective, is that the benefits of Christ's work can largely be appropriated by faith in this present age as:
Christ redeemed us from the curse of the law by becoming a curse for us, for it is written: “Cursed is everyone who is hung on a pole.” - Galatians 3:13 NIV
“He himself bore our sins” in his body on the cross, so that we might die to sins and live for righteousness; “by his wounds you have been healed.” - 1 Peter 2:24 NIV
Note that 'redeemed' and 'healed' are past tense, establishing a basis for belief that Christ's words on the cross of 'It is finished' can legitimately be given a broad application that would cover deliverance from all aspects of the curse, including the pains of childbirth. Under this framework, the Saints of God no longer 'die', but merely go 'to sleep' to awaken at the last trumpet.
Further, although it is part of a difficult passage to interpret, women are given hope that:
But she will be delivered through childbearing, if she continues in faith and love and holiness with self-control. 1 Timothy 2:15 NET
Could well include deliverance not just from death, but also the pain of the curse. If some regard that as a fanciful application of this particular scripture, perhaps they will be more reticent to deny that:
Everything is possible for one who believes. - Mark 9:23b NIV
Can be a legitimate basis for such a belief. In any event, there are many that have dared to believe such a thing and have subsequently testified to encountering God's deliverance in this manner.
PS Your talk of the Old Testament and shrimps is a bit of a red herring with regard to the substance of this question, but the point should be made that Jesus explicitly did not 'revoke' a 'bunch of other stuff in the Old Testament' - He came to fulfill the law perfectly. That laws expressly given to mark Israel, and subsequently the Jews, apart as the Covenant people the Messiah would come through were not mandated holus-bolus on Gentile converts does not constitute a 'revocation' and each law in the Old Testament should be understood in it's proper context and evaluated for utility as a guiding principle (not moral obligation that brings condemnation and judgment by it's transgression) in the life of a New Covenant believer. If Gentile converts had by and large followed the Jewish dietary laws (as a guiding principle for healthy living), particularly with regard to eating pork and shellfish, no doubt countless thousands of them would have experienced consequent benefit in extended and overall more healthy lives.