I think the OP is right. Owen is being uncharacteristically sloppy in his argument. But the result is really the same... Christ died only for his elect.
Owen is often pretty hard to understand, and often needs to be read four or five times before the penny drops. Once you have understood him you then have to try and work out if he is being Biblical or not. He is a bit of a slog and not recommended when you are tired, easily distracted, or tipsy.
Permit me to guess what Owen actually means by a "Universalist". What we mean today is a person who believes that literally all will be saved by the work of Christ, whether they believe in Christ or not, even if they have never heard of Christ all the days of their life. Owen is not using this definition of Universalist.
By "Universalist" Owen means those Christians who believe that the death of Christ was in every way you care to consider effective and intended for all men so that it truly makes possible the salvation of anyone: God has done all that he is able to do by giving his Son, and now He is waiting (helplessly as it were) for men to respond to His love in Christ.
In other words, most Christians today who are not Calvinist are "Universalists" using Owen's 17th century definition.
With this definition we can now consider what Owen is saying.
To get the force of his argument it might be better to consider the whole quote rather than the trnucated form in the question:-
God imposed his wrath due unto, and Christ underwent the pains of hell for
1) either all the sins of all men,
2) all the sins of some men,
3) some sins of all men.
If the last (3), some sins of all men, then all men have some sins to answer for, and so shall no man be saved; for if God enter into judgment with us, though it were with all mankind for one sin, no flesh should be justified in his sight: “If the LORD should mark iniquities, who should stand?” Ps. 130:3. We might all go to cast all that we have “to the moles and to the bats, to go into the clefts of the rocks, and into the tops of the ragged rocks, for fear of the LORD, and for the glory of his majesty,” Isa. 2:20, 21.
If the second (2), that is it which we affirm, that Christ in their stead and room suffered for all the sins of all the elect in the world.
If the first (1), (all the sins of all men) why, then, are not all freed from the punishment of all their sins? You will say, “Because of their unbelief; they will not believe.” But this unbelief, is it a sin, or not? If it is not sin, why should they be punished for it? If it is sin, then either Christ underwent the punishment due to it or he did not. If he did, then why must that hinder them from partaking of the fruit of his death more than their other sins for which he died? If he did not undergo the punishment do to the sin of unbelief, then did he not die for all their sins.
Let them choose which part they will.
Owen himself believes in option 2 - that Christ died for all the sins of some men, i.e. the elect.
What Owen is arguing is that the other options, 1 and 3, are untenable.
So what Owen is saying is that the Christian who is not a Calvinist (i.e. the typical Christian today) has got a problem: if Christ died for all the sins of all men then why are not all saved? The typical believer today will answer "Because of unbelief, they do not trust in Christ", to which Owen argues "But unbelief is a sin, and you are saying Christ died for all the sins of all men (otherwise the problem of option 3 arises), so why is there still a punishment for this sin (of unbelief), when you are saying that Christ died for it (because you claim Christ died for all their sins)?"
The OP's question is justified, there appears to be an inconsistency in Owen's line of argument, even though in the final result it makes no difference. The elect are not saved without any faith: but the elect have not saving faith the second before they are saved, and they are forgiven that unbelief and saved by being given faith from above. And why are they forgiven for their previous unbelief? Because Christ paid the penalty for that sin and all the sins of the elect.
What Owen should have said was that if Christ died for all the sins of all people then that would include the unbelief of all people which means there must then be nothing in the way to prevent God fully forgiving everyone and giving everyone faith and salvation, in just the same way as he gives faith to his elect despite their previous unbelief.
Is salvation by faith? Yes.
Did Christ die for all sins of all people? NO!! says Owen.
Are all saved? No.
Why? Ultimately, because Christ did not die for all, but only for His elect.
How do we know this? Because if lack of faith in every individual is one of the sins for which Christ died, then every individual should come to faith and salvation. If everyone's unbelief is paid for by the death of Christ then their unbelief will not disqualify them and everyone of necessity will come to faith and salvation. The fact that multitudes never come to Christ is proof that their sins were never paid for in the first place.
If I live in unbelief for the first 20 years of my life then nothing I do can pay for that sin, it is a sin worthy of everlasting condemnation. But God gives me faith and salvation because Christ took the punishment for that unbelief on the cross. But if Christ took the punishment for that unbelief for all men then there is no reason why God should not give faith and salvation to all men.