Could you tell me how it is that Jesus “died for our infirmities and diseases”, yet we certainly do not lack these?
This quotation in Matthew 8:17 comes from Isaiah 53:4. Matthew claims its "fulfillment" in that Jesus is seen in context healing many people with physical diseases and some with supernaturally induced ones as well (resulting from demon possession). We should note right away that although our Lord's healing ministry was more widespread and impressive than any other before or since, He did not heal everyone, not even everyone in Israel, not even everyone who put their faith in Him, and even those whom He did heal were not thereafter immune to other diseases. So not only does this verse not speak to any universal furloughing of believers from diseases, it also cannot be taken to mean this even in Jesus' day. This is not to diminish our Lord's ministry - He is God and could certainly have removed all disease from all people for all time. The miracles had a purpose, in part to alleviate some suffering, but the larger purpose of leading to salvation those willing to come was clearly far more important than any temporary relief from pain.
For those familiar with the context of the quote, Isaiah 52-53, none of this is surprising, because that passage is primarily concerned with the suffering of the Servant whose sacrifice redeems us from our sins. Matthew quotes this verse because Jesus' ministry of healing not only marks Him out as this Servant, the true Messiah (cf. our Lord's comments to John's disciples at Matt.11:4-6), but also is symbolic of the true fulfillment of this verse in Jesus' bearing of our sins on the cross. In other words, the partial, symbolic fulfillment of Is.53:4 on this occasion noted by Matthew looked forward to the near future fulfillment at Calvary (cf. Rom.4:25). This is clear enough from the quotation from 1st Peter below, where Peter clearly has Isaiah 53:3-5 in mind:
He Himself bore our sins in His body on the tree, in order that we might die to sins and live to righteousness. By His wound you are healed.
1st Peter 2:24
This connecting of sin with pain and disease is a natural one for anyone familiar with scripture to make. Disease can be a direct consequence of personal sin (cf. the communion abuses at Corinth: 1Cor.11:29-30, and Jesus command to the just healed paralytic "go and sin no more": Jn.5:14). But it is not necessarily even the rule, for there is also a sense in which all infirmity and pain are a result of Adam's sin through which we have come to inhabit these imperfect bodies of sin, bodies that since the expulsion of our parents from Eden are mortal and subject to all manner of pain and disease through what have become natural processes.
Therefore in scripture disease is a natural and obvious metaphor for sin, since disease is sometimes a result of sin, and without original sin there would be no disease at all. So it is not surprising that one finds disease standing for sin frequently in Bible (cf. Ps.38:17-18; 69:26). We should understand both the language of Isaiah and the quotation in Matthew as they apply to Jesus in precisely these terms. That is to say, both "He hath borne our griefs and carried our sorrows" (Is.53:4 KJV) and "He Himself took our infirmities and bore our sicknesses" (Matt.8:17 KJV) have their primary application in Jesus' work on the cross, and neither implies that we have been relieved from sickness in this world while we continue to reside in these bodies of sin.
It is important to note that in the verse in Isaiah which precedes the one quoted in Matthew the exact same Hebrew words are used of the Servant's own personal suffering (machobh and choli - KJV does a good job on this). Since without question our Savior was without personal sin, the statement that He would be "a Man of sorrows, acquainted with grief" has to refer to all the physical and mental pain Jesus endured during His earthly life in spite of the fact that His body was free of sin. We are reminded of Paul's references to sharing the sufferings of Christ (Col.1:24; cf. Rom.8:17; 2Cor.1:5; 4:10; Gal.6:17; Phil.3:10; and Peter too: 1Pet.4:13), and this surely must refer to His life of sacrifice (since we cannot and need not die to propitiate sin as He did). We can say of a certainty that our Lord's life entailed much pain. Not only did He live a live of physical hardship and mental anguish (Heb.12:3), He had to run the ultimate gauntlet of trials and torture just to get to the cross where He could then be forsaken and judged for all of our sins.
This question has much in common with the "why are we still here" one. In prayer in Gethsemane, Jesus asked not that we be taken out of this world, but that we be sanctified in it by means of the truth (Jn.17:15-17). This was so that we might continue in life to serve and glorify Him through spiritual advance and to help our fellow Christians advance as well. Disease is much like any other category of opposition we face in the devil's world - if there were no opposition, then our free will in choosing for Jesus day by day and moment by moment would not be receiving a fair test (and there really would be no need for us to stay here past salvation). But we find ourselves in the middle of a seven millennia long struggle between the Lord and the evil one, and we have to play our part according to the rules, suffering and disease being some of the "flak" we have to fly through on this mission; without challenge, struggle and opposition, there would not only be no reason for us to continue in life after accepting Jesus, there also would be no basis for the rewards we anticipate and hold dear, for they are based upon our performance in the ranks under the difficult circumstances of this earthly life.
Our Lord knew pain and suffering - even before the cross, and so must we His servants. There are many varieties of opposition, sorrow, disease and the like (cf. Paul's considering of his concern for the churches as in some respects more onerous than anything else: 2Cor.11:28), but in Jesus there is hope, there is joy, and there is ultimate deliverance of which we are certain. And on top of that, Jesus has left us His peace (Jn.14:27), so that no matter what we are called upon to endure, we can rest easy in the warm embrace of our Savior, waiting for the day of our ultimate triumph. In the meantime, we are called to remember His example of unflagging courage in the face of suffering, pain and opposition, and do our best to follow in His footsteps (1Pet.2:21-23).
For more on the nature of sin and our Lord's death and sacrifice to redeem us from it, see Bible Basics 3B: Hamartiology: the Biblical Study of Sin. And now please also see in part 4A of Basics, Christology, "The Spiritual Death of Christ".
In the One who carried all our infirmities and who intercedes for us before the throne of grace, our Savior Jesus Christ.