This is false. John was not writing to Israel in 2 John. He was writing to the "elect lady" and the "children" of her assembly. These "children" were not her own bodily offspring, any more than John addressing the church members he was teaching as "my little children" was speaking to natural-born sons and daughters of John.
This "elect lady" (who was responsible for either admitting or refusing membership for those wishing to join her assembly) also had an "elect sister" leading another congregation. John was passing on greetings from one woman's assembly to another woman's assembly.
None of this disagrees with Paul, who laid out instructions for how Timothy was to pass on exhortation to elders of the flock of either gender, whether young or old, male or female in 1 Timothy 5:1-2.
There is nothing in 2 John that identifies the elect lady as being a local assembly
Who is elect?
[Isa 42:1 KJV] 1 Behold my servant, whom I uphold; mine elect, [in whom] my soul delighteth; I have put my spirit upon him: he shall bring forth judgment to the Gentiles.
Jesus is elect
[Isa 45:4 KJV] 4 For Jacob my servant's sake, and
Israel mine elect, I have even called thee by thy name: I have surnamed thee, though thou hast not known me.
Israel is elect
[Col 3:12 KJV] 12 Put on therefore, as the elect of God, holy and beloved, bowels of mercies, kindness, humbleness of mind, meekness, longsuffering;
The church the body of Christ is elect as it it chosen in Christ
[1Ti 5:21 KJV] 21 I charge [thee] before God, and the Lord Jesus Christ, and the elect angels, that thou observe these things without preferring one before another, doing nothing by partiality.
Angels are elect
[2Jo 1:6 KJV] 6 And this is love, that we walk after his commandments. This is the commandment, That, as ye have heard from the beginning, ye should walk in it.
Israel heard from the beginning to walk in love
[Jhn 14:15 KJV] 15 If ye love me, keep my commandments.
This was directed to Israel
[Jer 3:6 KJV] 6 The LORD said also unto me in the days of Josiah the king, Hast thou seen [that] which backsliding Israel hath done? she is gone up upon every high mountain and under every green tree, and there hath played the harlot.
[Amo 5:2 KJV] 2 The virgin of Israel is fallen; she shall no more rise: she is forsaken upon her land; [there is] none to raise her up.
Israel is referred to as a lady
I think it more likely that believing Israel is the elect lady but I am not dogmatic in this