Generally, there are both practical and spiritual reasons for the choices that God makes.
God, contrary to the opinions of unbelievers, is fair and reasonable. There would be both rich and poor among the children of Israel, and the Lord wanted everyone to be able to make offerings. So He laid out for them the kinds of creatures that could be sacrificed. Notice, if they were to picture Christ, they all must be ceremonially clean animals; chapter 11 will explain that for us. Later, in chapter 22, we will have a careful description of any flaws that would disqualify an animal from being offered. In a word, “It must be perfect to be accepted” (Lev 22:21). But in Leviticus 1 we have the various groups that could be offered. The three categories were those from the herd, from the flocks, or from the birds. May I suggest the young bull or ox pictures Christ’s unwavering service. These beasts of burden were utilized for plowing (1 Sam 14:14, KJV), for treading out corn (Deut 25:4), and as draft animals (Num 7:3). Only seldom were they killed for meat. Those from the flock emphasize His uncomplaining suffering. A lamb has no claws or fangs or horns to protect it. Thus our amazement in the book of the Revelation when God sends out His little Lamb to stand against the ferocious beasts spawned in hell. But the Lamb had already won, when once it went “as a lamb to the slaughter, and as a sheep before its shearers is silent, so He opened not His mouth” (Isa 53:7). The little birds, able to be caught by the poor, remind us of Christ’s unquestioning submission (Gen 8:10; Job 38:41; Lk 12:24), as He yielded to the dove-like Spirit. But what humility! Can you imagine being compared to a dead goat or a beheaded bird? Yet God was willing to go this far if it would help us grasp the sacrifice of His Son.