Paul is referring in this
1 Timothy 5:1-2 verses to elders in the ministry of either male or female gender, both younger and older in years.
There are other Greek words which Paul could have used if he was strictly addressing only the age factor. Paul could have used the same word for an "older man" - a "
presbutes" - that Zechariah, John's father called himself in
Luke 1:18. "...I am an old man (
presbutes), and my wife is well stricken in years." Or Paul could have used "
presbytas", which refers to an aged-in-years man in
Titus 2:2. "That the aged men (
presbytas) be sober, grave, temperate, sound in faith, in charity, in patience." Or he could have used "
presbytidas", meaning an aged-in-years woman in
Titus 2:3. "The aged women (
presbytidas) likewise, that they be in behavior as becometh holiness, not false accusers..."
But Paul didn't use these terms in
1 Timothy 5:1-2. He used "presbutero" and "presbuteras" instead, which carries the meaning of these being
overseers in the assembly - not just ordinary senior citizens in the church.
The "elect lady" (kyria) was a specific woman that John expressed a desire to see "face to face". Her "children" were the members of the assembly that this "elect lady" was leading. John was giving this woman instructions on just who she was to admit to the fellowship and who should not be allowed entry into the assembly, if they did not follow the doctrine of Christ. This was protective shepherding that this woman was being coached in by John. She also had a fellow "sister" leading another assembly of "children", for whom John passed on greetings.
Since John already was referring to the members of the church body as "children" in this epistle, it would have been redundant to also refer to the "elect lady" as being the church, with the church as "children" under that church's leadership.