Refutation of Chizzuk Emunah, Part II, Ch. XII: "Doctrine of the Divinity of Jesus Examined"

My first reaction is that this is simply not the accepted understanding of the passage, and that this understanding can only come from a person determined to find fault in it. But that hardly seems to do it justice, so here's a better, more complete answer:

One obvious response to the idea that Matthew 8:20 refers to Jesus acknowledging that He isn't divine is that it's odd that a Jew would not know their own Scripture. Although the phrase "Son of Man" was a phrase of "humilation and debas*m*nt", the phrase actually points to His divinity, as the foretold one, right from the book of Daniel 7:13(KJV)

I saw in the night visions, and, behold, one like the Son of man came with the clouds of heaven, and came to the Ancient of days, and they brought him near before him.

Moreso, the fact that Jesus would come in humility is right out of Psalm 8:5.

For thou hast made him a little lower than the angels, and hast crowned him with glory and honour.

As for why Jesus would want to discourage this person from following Him, a common understanding is that this scribe expected material gain, and Jesus was just giving him a realiztic expectation of the life he would lead if he followed Christ.

From Gill's Exposition of the Entire Bible

...So that though he was Lord of all, as being the mighty God; yet as "the son of man", a phrase, expressive both of the truth and meanness of his human nature, the most despicable of creatures in the earth and air, were richer than he. This he said, to convince the Scribe of his mistake; who expected much worldly grandeur and wealth, by becoming his disciple.

As a matter of fact, Gills Exposition refutes he entire argument put forth by Yitzchak ben Avraham

When Christ styles himself "the son of man", it is no contradiction to his being God; nor any objection to trust and confidence in him, as the Jew (z) suggests; for he is truly and properly God, as well as really man, having two natures, human and divine, united in his person; so that he is, as was prophesied of him, Emmanuel, God with us, in our nature, God manifested in the flesh: and since he is so, it cannot be unlawful to trust in him; which it would be indeed, was he a mere man. The Jews ought not to object to this name and title of the "Messiah, the son of man": since he is so called, as their own writers and commentators acknowledge, in (a) Psalm 80:17 and (b) Daniel 7:13. And whereas it is further urged against these words of Christ, that if he was God, why does he complain of want of place? Is not the whole world his, according to Psalm 24:1? It may be replied, that it is very true, that the whole world is his, nor could he be in want of anything, as God; but yet, as man, for our sakes he became "poor", that we "might be rich": nor should this be any difficulty with a Jew, when they themselves say, as some have thought, if he (the Messiah) should come, , "there's no place in which he can sit down" (c). Unless it be understood of Nebuchadnezzar, as the gloss explains it; let the learned inspect the place, and judge: the coming of the Messiah is immediately spoken of.

As does Matthew Henry's Concise Commentary

8:18-22 One of the scribes was too hasty in promising; he proffers himself to be a close follower of Christ. He seems to be very resolute. Many resolutions for religion are produced by sudden conviction, and taken up without due consideration; these come to nothing. When this scribe offered to follow Christ, one would think he should have been encouraged; one scribe might do more credit and service than twelve fishermen; but Christ saw his heart, and answered to its thoughts, and therein teaches all how to come to Christ. His resolve seems to have been from a worldly, covetous principle; but Christ had not a place to lay his head on, and if he follows him, he must not expect to fare better than he fared. We have reason to think this scribe went away. Another was too slow. Delay in doing is as bad on the one hand, as hastiness in resolving is on the other. He asked leave to attend his father to his grave, and then he would be at Christ's service. This seemed reasonable, yet it was not right. He had not true zeal for the work. Burying the dead, especially a dead father, is a good work, but it is not thy work at this time. If Christ requires our service, affection even for the nearest and dearest relatives, and for things otherwise our duty, must give way. An unwilling mind never wants an excuse. Jesus said to him, Follow me; and, no doubt, power went with this word to him as to others; he did follow Christ, and cleaved to him. The scribe said, I will follow thee; to this man Christ said, Follow me; comparing them together, it shows that we are brought to Christ by the force of his call to us, Ro 9:16.

The idea that Jesus "knew He wasn't God" doesn't hold water in light of other Scripture, either. It would be laughable if it weren't so sad. The question "Did Jesus claim to be God?" has been answered and addressed so thoroughly, and is such an established teaching, that it's simply ridiculous to try to claim otherwise. One example here.

The whole argument ignores the basic orthodox Christian teaching that Jesus assumed a humble nature willingly, for His purpose, but knew full well of His divinity.