I think the sentence has a natural meaning which makes good sense here. To bring something to light means to make it visible. The dawning sun brings a landscape to light. A good maths teacher by careful explanation can bring to light the principles required to solve maths equations. An activist brings to light social justice issues that other people were not previously aware of.
In the same way Jesus has brought to light life and immortality through the gospel. The gospel is the story that Jesus is the incarnation of God; that he died and rose again; and that in union with Christ others can share in that resurrection life. Before Jesus all this was hidden from sight; now it is visible because of Jesus and because of the testimony of his apostles.
Paul writes about this theme in other ways and in other letters. For example in Colossians 1 he describes "the mystery that has been kept hidden for ages and generations, but is now disclosed to the Lord’s people." (Colossians 1.26; see also Ephesians 3.) This is not a mystery in the sense of being hard to understand; it's a mystery in the sense of being a secret that is hidden. It was once known only to God, but now it is known also to God's apostles and by extension to the church.
So also, Peter talks of the living hope that we have received through the resurrection of Jesus Christ (1 Peter 1.3). It is a hope that we have because we are experiencing the very salvation of God in Jesus Christ. So we know about it and testify to it, says Peter. Yet this salvation was a mystery to the prophets of old, and even the angels themselves, who searched for and longed to understand these things. (1 Peter 1.10-12)
In summary, we are simply talking of God revealing things now which in the past were not known or understood. If this is correct then 2 Timothy 1.10 has no clear connection with 1 Timothy 6.16. The metaphor of light is suggested in both texts; but note that in the Greek we are comparing a noun (φῶς) and a verb (φωτίσαντος). So It's not really a parallel between light and light. In English we would get a better sense of the difference if we compared "light" and "enlighten". In 2 Timothy 1 it's not about light; it's about the phrase "brought to light", which is simply another way of describing something made known.