What about 1 Timothy 4:10? If Jesus doesn't actually save all men, then he is not the savior of all men, right?
1 Timothy 4
English Standard Version
Some Will Depart from the Faith
4 Now the Spirit expressly says that in later times some will depart from the faith by devoting themselves to deceitful spirits and teachings of demons, 2 through the insincerity of liars whose consciences are seared, 3 who forbid marriage and require abstinence from foods that God created to be received with thanksgiving by those who believe and know the truth. 4 For everything created by God is good, and nothing is to be rejected if it is received with thanksgiving, 5 for it is made holy by the word of God and prayer.
A Good Servant of Christ Jesus
6 If you put these things before the brothers,[a] you will be a good servant of Christ Jesus, being trained in the words of the faith and of the good doctrine that you have followed. 7 Have nothing to do with irreverent, silly myths. Rather train yourself for godliness; 8 for while bodily training is of some value, godliness is of value in every way, as it holds promise for the present life and also for the life to come. 9 The saying is trustworthy and deserving of full acceptance. 10 For to this end we toil and strive, because we have our hope set on the living God, who is the Savior of all people, especially of those who believe.
11 Command and teach these things. 12 Let no one despise you for your youth, but set the believers an example in speech, in conduct, in love, in faith, in purity. 13 Until I come, devote yourself to the public reading of Scripture, to exhortation, to teaching. 14 Do not neglect the gift you have, which was given you by prophecy when the council of elders laid their hands on you. 15 Practice these things, immerse yourself in them,[c] so that all may see your progress. 16 Keep a close watch on yourself and on the teaching. Persist in this, for by so doing you will save both yourself and your hearers.
Great question. In one sense (a), He is the Savior of all men, and died for all, but in another (b), He is not and did not:
(a) He is the only one through Whom we or anyone can be reconciled to the Father; this is the
general, outward call of the gospel and Scripture, to repent and believe. Jesus is the only Savior, and there is no other way to the Father except through Him. And, His sacrifice is
sufficient for all.
(b) The
specific, inward call is only issued ~ by God through the work of His Holy Spirit, each at his/her God-appointed time ~ to those whom God chose before the foundation of the world (Ephesians 1:1-14, especially verse 4) to be conformed to Christ Jesus. These are God's Elect, His true Israel, and a subset of all humanity. These are the ones for whom Jesus's sacrifice was
efficacious.
To speak specifically to what you quoted, we could substitute the word
'effectually' for
'especially' in 1 Timothy 4:10. In John 6, Jesus says,
"All that the Father gives Me will come to Me, and whoever comes to Me I will never cast out. For I have come down from heaven, not to do My own will but the will of Him Who sent Me. And this is the will of Him who sent Me, that I should lose nothing of all that He has given Me, but raise it up on the last day."
In what Jesus says here, there is the strong implication that the Father did not give Him all human beings, but only some/many. So, not all are members (or destined to be members) of God's Elect. As Jesus says in John 5:
"...an hour is coming when all who are in the tombs will hear (the Father's) voice and come out, those who have done good to the resurrection of life, and those who have done evil to the resurrection of judgment."
Later, in John 10, Jesus says:
"My sheep hear my voice, and I know them, and they follow me. I give them eternal life, and they will never perish, and no one will snatch them out of My hand. My Father, Who has given them to Me, is greater than all, and no one is able to snatch them out of the Father’s hand. I and the Father are one."
Again, the strong implication is that not all are His sheep.
I welcome your thoughts. Grace and peace to you.