1) The Apostles know how to interpret the words of Jesus better than we. In their teaching and letters, they viewed themselves as spiritual fathers, and as having spiritual sons (both which implies the other).
1 Corinthians 4:14-17 (DRB) I write not these things to confound you; but I admonish you as my dearest children. For if you have ten thousand instructors in Christ, yet not many fathers. For in Christ Jesus, by the gospel, I have begotten you. Wherefore I beseech you, be ye followers of me, as I also am of Christ. For this cause have I sent to you Timothy, who is my dearest son and faithful in the Lord; who will put you in mind of my ways, which are in Christ Jesus; as I teach every where in every church.
2 Timothy 1:2 (DRB) To Timothy my dearly beloved son, grace, mercy, and peace, from God the Father, and from Christ Jesus our Lord.
1 Peter 5:13 (DRB) The church that is in Babylon, elected together with you, saluteth you: and so doth my son Mark.
Male persons who identify people as their sons have, by virtue of this identification, identified themselves as fathers.
Matthew 3:9 (DRB) And think not to say within yourselves, We have Abraham for our father. For I tell you that God is able of these stones to raise up children to Abraham.
Cf. Gn. 17:5.
Here Abraham is described as having spiritual sons (believing Jews and Gentiles, without respect to their geneology), as in:
Galatians 3:7 (DRB) Know ye therefore, that they who are of faith, the same are the children of Abraham.
Luke 16:19-25 (DRB) There was a certain rich man, who was clothed in purple and fine linen; and feasted sumptuously every day. 20 And there was a certain beggar, named Lazarus, who lay at his gate, full of sores, 21 Desiring to be filled with the crumbs that fell from the rich man's table, and no one did give him; moreover the dogs came, and licked his sores. 22 And it came to pass, that the beggar died, and was carried by the angels into Abraham's bosom. And the rich man also died: and he was buried in hell. 23 And lifting up his eyes when he was in torments, he saw Abraham afar off, and Lazarus in his bosom: 24 And he cried, and said: Father Abraham, have mercy on me, and send Lazarus, that he may dip the tip of his finger in water, to cool my tongue: for I am tormented in this flame. 25 And Abraham said to him: Son, remember that thou didst receive good things in thy lifetime, and likewise Lazareth evil things, but now he is comforted; and thou art tormented.
3) God doesn't interpret this teaching as excluding spiritual fathers.
Isaiah 22:20-23 (DRB) And it shall come to pass in that day, that I will call my servant Eliacim the son of Helcias, 21 And I will clothe him with thy robe, and will strengthen him with thy girdle, and will give thy power into his hand: and he shall be as a father to the inhabitants of Jerusalem, and to the house of Juda. 22 And I will lay the key of the house of David upon his shoulder: and he shall open, and none shall shut: and he shall shut, and none shall open. 23 And I will fasten him as a peg in a sure place, and he shall be for a throne of glory to the house of his father.
Cf. Mt. 16:13-20.
Pope (Papa) means 'father' and if this basically approves of a papacy-like position within the kingdom, wherein he functions as some kind of father ("he shall be as a father to the inhabitants of Jerusalem"), then the Bible can't be forbidding the idea.
4) It cannot be illict to identify someone as what the Bible says they are—fathers, or sons. It would be contrary to reason to think Jesus is teaching us to not call people what they are in truth. That is, it's not rational to hold the view that Jesus and others call people sons and themselves fathers, and obviously in an approved capacity, yet is against being truthful in this regard when identifying them.
5) In context, He is very clearly using hyperbole (as He does a lot) in order to drive a point through; He doesn't literally means 'don't call anyone teacher' or 'father.' And before you say, 'why didn't He teach it another way,' that's what hyperbole is: unqualified extreme.
Matthew 5:30 (DRB) And if thy right hand scandalize thee, cut it off, and cast it from thee: for it is expedient for thee that one of thy members should perish, rather than that thy whole body be cast into hell.
Here again, He speaks hyperbolically, with not the least qualification. There you have it, Jesus teaches perpetual dismemberment until you have no arms or eyes or legs, etc. (Notice in the passage we're discussing he doesn't qualify 'anyone.' That excludes even biological fathers if taken in the absolute.)
Much like metaphor, which is also unqualifed by nature: "God is a man of war;" (Ex. 15:3) "God is not a man" (Num. 23:19). Clearly He meant both, but one was metaphorical and the other doesn't qualify for the same given the context—taken with a little seasoning of common sense.
Matthew 23:1-12 (DRB) Then Jesus spoke to the multitudes and to his disciples, 2 Saying: The scribes and the Pharisees have sitten on the chair of Moses. 3 All things therefore whatsoever they shall say to you, observe and do: but according to their works do ye not; for they say, and do not. 4 For they bind heavy and insupportable burdens, and lay them on men's shoulders; but with a finger of their own they will not move them. 5 And all their works they do for to be seen of men. For they make their phylacteries broad, and enlarge their fringes. 6 And they love the first places at feasts, and the first chairs in the synagogues, 7 And salutations in the market place, and to be called by men, Rabbi.
To break here for a little:
These hypocritical leaders use their position of power to gratify their own pride. That's why they love being called 'Rabbi'—which is quite evidently equivalent to the modern day, 'Father.' He is very specifically talking about them.
The change to the more general seems to put most people off, and they lose sight of the hyperbole clearly being employed in the following:
8 But be not you called Rabbi. For one is your master; and all you are brethren. 9 And call none your father upon earth; for one is your father, who is in heaven. 10 Neither be ye called masters; for one is your master, Christ.
Does anyone really think that because God is father, there are no spiritual fathers, as St. Paul says? Does anyone really think that since Christ is our Master, there are no kings, lords, etc. "to whom honor is due" (Rom. 13:7)?
1 Peter 2:15-18 (DRB) For so is the will of God, that by doing well you may put to silence the ignorance of foolish men: 16 As free, and not as making liberty a cloak for malice, but as the servants δουλοι of God. 17 Honour all men. Love the brotherhood. Fear God. Honour the king.
18 Servants, be subject to your masters [δεσποταις] with all fear, not only to the good and gentle, but also to the froward.
Jude 1:4 (DRB) For certain men are secretly entered in, (who were written of long ago unto this judgment,) ungodly men, turning the grace of our Lord God into riotousness, and denying the only sovereign Ruler [δεσποτην], and our Lord Jesus Christ.
Clearly the Apostles didn't view the Lordship of Christ as exclusive to earthly lords, any more than they viewed God the Father as exclusive of spiritual fathers.
Jesus is just criticizing the titles-for-teachers practice as abused by prideful men—almost to the point of abolishing it in despair due to its woeful misuse and abuse—yet we see that He did not abolish the practice, or I should say perennial human tradition, of calling teachers by honorifics.
But what follows affirms what I said just before:
11 He that is the greatest among you shall be your servant. 12 And whosoever shall exalt himself shall be humbled: and he that shall humble himself shall be exalted.
It is the titles qua titles which occasion pride in prideful men which are forbidden—if sought after and 'enjoyed'—not the simple titles themselves.
Taken in a literal, anti-contextual way, this would forbid all honorifics whatsoever, since 'only God deserves to be honored.' Yet this is evidently not what is meant, given the Biblical data.
2 John 1:1 (DRB) The ancient to the lady [κυρια] Elect, and her children, whom I love in the truth, and not I only, but also all they that have known the truth,