Well, the bible is the sword of the spirit, the word of God and if it does not correct a person it will cut him and cause him to turn from it. It has to be approached humbly and prayfully recognizing that the reader does not know better then its author. Yes, it is good to ask these types of questions since this is how we learn and verify our understanding. But we must be willing to challenge our beliefs and diligently seek out a correct understanding that harmonizes with the entire bible.
This scripture is a great example of how our ways are not Gods. Even if we believe we are doing something good if it is not according to how God instructs us then we are not doing his will. We are not following him (I Sam15:11) and actually doing evil in His sight (v19). This is what Saul did by sparing Agag and the best of the sheep and oxen. His action did not align with Gods command in verse 3.
In answer to your question. This passage also helps us understand that as the Bible says there are non-righteous (Rom 3:10), we are all gone astray (Ecc 7:20) and are wicked from the womb (Ps 58:3). This is why God, by His mercy and grace (Eph 2:8) had to chose those He intended to save (Eph 1:4-13) and why we cannot do anything to contribute to our salvation. Salvation is 100% in Gods hands and He is perfectly just.
No one in Amalek was chosen to be saved. Children even new born infants are sinful. Just like only Noah and his family was spared during the flood. Like in Sodom and Gomorrah, only Lot and his children were saved from the destruction (even his wife was destroyed). Like one thief was saved on the cross or like Jacob was loved by God but Esau was not (Ro 9:13). The fact is that God hates sin and the sinner (Psalm 5:5, Psalm 11:5), and we are all guilty of sin from the womb.
We have to let God be true (Rom 3:4). This is why we have to constantly test our ways to make sure we are on a correct path lighted by the gospel and not by our sin tainted mind and compassionate feelings. The bible is Gods Word, His law, the sword of the Spirit. We have to prayfully study it and pray God to grant us correct understanding and the strength to do His will. This is particularly important today as there is an explosion of false teachings in the world as God warned in the Bible (Matt 24). If we do not do exactly as God commands then we are doing evil in his sight just as Saul did. There is only one true gospel (Gal 1:9, Eph 4:14) are we going to submit to it no matter how much we think it just does not seem right or are we just going to tweak it a little so it is more agreeable with us, which we are also warned about in the Bible (2 Tim 4:3). We live in a sin cursed world and the Gospel, the bible, which is of God and not of this world, should cause us to challenge our beliefs and result in struggles through which God will correct us if it be His will to do so.
The bible (or Gods commands) is very hard to succumb to. After all how can a baby be guilty of sin? How can judgement start in the house of God (I Peter 4:17) and why will Christ turn away so many that believed they were doing good works (Matt 7:23, Matt 25:41, Luke 13:27, Matt 7:13). But it is Gods word (John 1:14). Christ is the loving, faithful savior of the elect and also the perfect judge. Are we going to listen and do what He commands or are we going to do what we think are better (Prov 21:2) like Saul, Jonah, David, Moses... Left up to us to choose righteousness we will fail every time. Our compassion, anger and pride will get in the way. Thank God for his mercy and for giving us His Word for without it we would remain a prisoner to our wicked hearts (Jeremiah 17:9, Mark 7:21-23, Is 64:6, I Thess 2:4).
This is why it so important to be in the environment in which God saves which is under the hearing of His Word (Romans 10:17 ). Even babies, deaf or mentally challenged people need to be in that environment. Thankfully salvation does not require understanding. That will follow salvation.