The problem is that that is not how temptation arises. Temptation occurs from within. James 1 states:
When tempted, no one should say, “God is tempting me.” For God cannot be tempted by evil, nor does He tempt anyone. But each one is tempted when by his own evil desires he is lured away and enticed. Then after desire has conceived, it gives birth to sin; and when sin is fully grown, it gives birth to death.
Scripture also says:
For we do not have a high priest who is unable to sympathize with our weaknesses, but we have one who was tempted in every way that we are, yet was without sin. (Hebrews 4:15)
Now, if Jesus really was tempted in EVERY way that we are, then He had to have those same "evil desires" described in James 1:14-16, from which our personal temptations arise. If He did not have this, then the Scripture in Hebrews is not really true. This is why we say He has the truly human nature.
If the flesh was really completely unimportant, as you seem to be implying, then physical sins and physical actions are really unimportant. They mean nothing. We cannot divorce the concept that Christ did not have a human nature from the concept that He did not have a human body. As Athanasius said, whatever was not assumed was not redeemed. If He did not assume the natural human nature, then He did not redeem it.
Overall, the claim that Jesus did not have a fully human nature runs into many issues with Scripture itself, not the least of which being the fact that it makes many Scriptures out to be lies. Monophysites tried to rationalize this away, and it is certainly a lot easier to rationalize the nature of Christ rather than to simply accept Him as He is, but we do not have that luxury. Anything that attacks the nature of Christ as being fully God and fully man is turned away as false. This is what we profess in the creed of Chalcedon:
We, then, following the holy Fathers, all with one consent, teach men to confess one and the same Son, our Lord Jesus Christ, the same perfect in Godhead and also perfect in manhood; truly God and truly man, of a reasonable soul and body; consubstantial with us according to the manhood; in all things like unto us, without sin; begotten before all ages of the Father according to the Godhead, and in these latter days, for us and for our salvation, born of the virgin Mary, the mother of God, according to the manhood; one and the same Christ, Son, Lord, Only-begotten, to be acknowledged in two natures, inconfusedly, unchangeably, indivisibly, inseparably; the distinction of natures being by no means taken away by the union, but rather the property of each nature being preserved, and concurring in one Person and one Subsistence, not parted or divided into two persons, but one and the same Son, and only begotten, God the Word, the Lord Jesus Christ, as the prophets from the beginning have declared concerning him, and the Lord Jesus Christ himself taught us, and the Creed of the holy Fathers has handed down to us.
Two natures. The nature of man does not necessitate sin. Sin is not the desire to do something. It is the enaction of that desire. Part of man's nature is free will, the ability to choose between the passions and the virtues. The desires can spring up within us to do either, whether evil or good. It is not until we give voice to those desires that it becomes sin. Christ had those same desires pop up in Him. One needs only look at His prayer at Gesthemene. He, in His human frailty, desired to escape the cross and death. But He did not allow that desire to grow and conceive and give birth to sin. He pushed it down and starved it, being without sin.
People act as if this is something that humans didn't have from the Creation, but the ability to choose necessitates this reality. One can choose to sin without having a sinful nature. Eve shows that same desire being within her as it was coaxed out by the serpent in the garden. But if the desire were not there, the serpent could not have coaxed it out. Have you ever tried to just talk to a computer and convince it to let you around an encryption on a file inside it? Doesn't really work that way, now does it? It doesn't have desires. You can't convince it to do something. The only way to convince someone to do something is to feed one of their desires. But without those desires, there is no temptation. So Adam had these desires from creation, as do all humans. The desires for the passions are not in and of themselves sinful. Think of it. With all of the passions, there are right and wrong uses:
Lust is sinful when it is directed at people in a damaging way that does not uplift them. When directed at people outside of marriage, in brief, fleeting one night stands, it is sinful. But when one comes together in the marriage bed with his wife, there is nothing sinful about it at all. There is nothing sad, or disgraceful when two married people have consensual, loving sex with each other. And if Mark Gungor is to be believed, it can be wildly passionate and even fun. But it is special. The reason God told us it was sinful outside of marriage was not because of some arbitrary rule to say "thou shalt not have fun". It was because He knew what the effect of sex outside of marriage would be, with numerous negative effects on people who engage in sexual intercourse without the commitment it is tied to.
Anger is sinful when it is out of control and not based in righteousness. There is a story of a monk who came to his abbot in the desert and asked, "what does it mean to be angry without a cause?" He was referring to a passage in the Scriptures commanding us to not be angry, and in some texts it says "without a cause". The abbot answered, "do not be angry with your brother, even if he were to dismember you, for he only causes damage to that which is temporary. Be angry rather, when you see a man trying to separate another from God." The meaning here is that those who cause physical harm to us personally cannot separate us from God. But when someone seeks to fool a person of weak spirit into following a heresy, or into worshiping another god, to cause that person to stumble, then we ought to be angry. Even Christ spoke of this kind of person with righteous anger, saying it would be best if the misleading person were to have a millstone hung from their neck and cast into the sea.
I could go on with an example for each of the seven passions, but we do not have that space. The point is that sin is not something innate, but something other than us. It is something other than what God intended, something that is not our birthright, nor is it something we are born with. It is a disease that we contract. If it weren't for the demons going around tempting people, it would be possible for a child to be raised without exposure to sin. Since we are "monkey see, monkey do" people, we do not come preloaded with the desire to sin. Whether it is from us seeing mom or dad do something and mimicking them, or by satan or his minions planting the thought in our heads, we would have no exposure to sin. The desire has to precede the action.