Commentaries on 1 Timothy 4:10

David Dickson (1583?-1663) An exposition of all St. Paul's epistles ... (source)(spelling modernized by me)

Vers. 10. For therefore we both labor, and suffer reproach, because we trust in the living God, who is the Savior of all men, especially of those that believe.

Reas. 4. Confirming the former Reasons. Because we suffer afflictions and straights, labors and reproach enough from unjust persecutors, for rejecting the inventions of men, and defending of true godliness, which consists in the exercises of Faith and Obedience, and those afflictions wee bear valiantly, from the hope of the promises, which are made to us that walk in this way of godliness: Therefore rejecting the inventions of men, follow after godliness.

Living] He amplifies this Reason, and confirms it from two properties in God, in whom we trust: The first is, God is the Living God, not only subjectively, or as to the subject Actuous, but also effectually, or in respect of us, he that gives life, who so performs his promises, that we know him to bee the author of truth and life. Secondly, God is the Savior of all men, as it is said, Psal. 36. Thou preservest man and beast, by his general goodness nourishing and supporting all men, making his Sun to rise upon the just and unjust: But chiefly and upon a more special ground, the Savior of believers, who relying upon his promises concerning the happiness of the life to come, renounce and reject humane inventions and opinions even in the dangers of persecution, and follow after the exercises of godliness.

Patrick Fairbairn The Pastor Epistles (1874) (source)

Bernard, John Henry – The Pastoral Epistles  in The Cambridge Greek New Testament, with Introduction and Notes  1922 (Source)

Bernard notes Wisdom 16:7.  Here is the KJV translation of that (not canonical) text:

Wisdom 16:5-10

5For when the horrible fierceness of beasts came upon these, and they perished with the stings of crooked serpents, thy wrath endured not for ever: 6But they were troubled for a small season, that they might be admonished, having a sign of salvation, to put them in remembrance of the commandment of thy law. 7For he that turned himself toward it was not saved by the thing that he saw, but by thee, that art the Saviour of all. 8And in this thou madest thine enemies confess, that it is thou who deliverest from all evil: 9For them the bitings of grasshoppers and flies killed, neither was there found any remedy for their life: for they were worthy to be punished by such. 10But thy sons not the very teeth of venomous dragons overcame: for thy mercy was ever by them, and healed them.

I note that "τὸν πάντων σωτῆρα" in Wisdom 16:7 is similar to, but not identical to, "σωτὴρ πάντων ἀνθρώπων" in 1 Timothy 4:10.