Why is Jesus portrayed and remembered in a state of suffering?

It seems your source of motivation is mainly related to crucifixes, which is where, in many modern Catholic churches, you can see Jesus depicted as dead.

To add a historical note on this regard, taken from the Catholic Encyclopedia, the custom of how Jesus is presented in crucifixes has changed over the centuries. I wrote an answer about this here. The key points are that, when the crucifix was first introduced into Christian art (around the sixth century), and until the thirteenth century, the depiction of Jesus in the cruficix was:

... the Crucified is shown adhering to the cross, not hanging forward from it; He is alive and shows no sign of physical suffering; ... The head is erect, and surrounded by a nimbus, and bears a royal crown. ... In a word, it is not Christ suffering, but Christ triumphing and glorious on the Cross.

In contrast, in the second period, which stretches until today:

... the head droops onto the breast (cf. Borgia, De Cruce Veliternâ, 191), the crown of thorns is introduced, the arms are bent back, the body is twisted, the face is wrung with agony, and blood flows from the wounds. In the thirteenth century complete realism is reached by the substitution of one nail in the feet, instead of two, as in the old tradition, and the resulting crossing of the legs. All this was done from artistic motives, to bring about a more moving and devotional pose. The living and triumphant Christ gives place to a Christ dead, in all the humiliation of His Passion, the agony of His death being even accentuated.

(emphasis mine).

I want to highlight here one phrase from the above quotation:

All this was done from artistic motives to bring about a more moving and devotional pose.

Whether that was the actual reason why the radical change in crucifixes over the centuries, personally, remembering Jesus suffering or dead in the cross is a much more powerful devotion-enhancing image than remembering Christ triumphant in the cross, which itself is a literal contradiction, as the cross, a humiliating death method for those times, is where He died.

Because, with St. Paul, "we preach Christ crucified," which, with a false understand (1 Corinthians 2:14) of the meaning of the suffering of Christ, or inded the idea of such on its face, is "foolishness" and "a stumbling block" (1 Corinthians 1:23).

We are to "proclaim the death of the Lord until He comes" (1 Corinthians 11:26; 1 Corinthians 2:2).

He is definitionally "portrayed as having been crucified": Galatians 3:1 (cf. John 20:27; Revelation 5:6).

[The Galatians could even have been shown the Shroud of Christ... thoughts]

Since the mission of Christ was to suffer, and it played the most serious element of His mission, it deserves depiction. This is not exclusive. There are statues of Christ not suffering also.

Catholics portray, worship, and promote Jesus in all of His states of being, actions, words, etc.

A strong emphasis is placed on his suffering and death on a cross because it was in this act that He atoned for the sins of all humanity. This is the single most important thing He did because without it, we have no hope.

Paragraph 623 of the Catholic Catechism states:

By his loving obedience to the Father, "unto death, even death on a cross" (Phil 2:8), Jesus fulfills the atoning mission (cf. Is 53:10) of the suffering Servant, who will "make many righteous; and he shall bear their iniquities" (Is 53:11; cf. Rom 5:19)

It would be highly mistaken to think that Catholics only focus on the suffering of Christ. The Catholic Church teaches that in rising from the dead He confirms who he is and the work that He accomplished by His death.

Paragraph 651 of the Catechism states:

"If Christ has not been raised, then our preaching is in vain and your faith is in vain."521 The Resurrection above all constitutes the confirmation of all Christ's works and teachings. All truths, even those most inaccessible to human reason, find their justification if Christ by his Resurrection has given the definitive proof of his divine authority, which he had promised.

I'm sure with a little investigation, many examples can be found of Catholic images, prayers, etc. the are devoted to the healthy, happy, resurrected, glorious, etc. natures of Jesus Christ.