Paul's referencing his arrival with a rod is not to be taken seriously. Paul would never beat anyone with a rod.
the church he had planted and cultivated the Corinthians church was his pride and joy in the kingdom his crown jewel. And what was going on was someone had openly fornicated with thier fathers wife and the others all were approving of it .
Paul was addressing thier sin and approval of sin . He told them this about the man
“For I verily, as absent in body, but present in spirit, have judged already, as though I were present, concerning him that hath so done this deed, In the name of our Lord Jesus Christ, when ye are gathered together, and my spirit, with the power of our Lord Jesus Christ, To deliver such an one unto Satan for the destruction of the flesh, that the spirit may be saved in the day of the Lord Jesus.
Your glorying is not good. Know ye not that a little leaven leaveneth the whole lump?”
1 Corinthians 5:3-6 KJV
the church was proud of what he had done Paul is telling them if they don’t repent and get right he would come with the same rod he decided against the man in reference who did this
“It is reported commonly that there is fornication among you, and such fornication as is not so much as named among the Gentiles, that one should have his father's wife.”
1 Corinthians 5:1 KJV
Paul’s warning was meant to be stern so they would repent as you say often in Paul’s writings as well as the gospel it’s about the message and not about a literal taking of a term like “ come with a rod “ he’s talking about coming in a stern nature and rebuking them for their sin of approving of what the man did among them
sin spreads when Christians approve of it , just like the camp of Israel and even the first generations in Noah’s day one man winning among the rest it woll
Spread like a wild fire or terrible cancer among believers
it is important also to show that this rebuke of that man hadn’t lasted forever
Paul wrote a second letter and addressed this matter which really shows the gospel
“But I determined this with myself, that I would not come again to you in heaviness. For if I make you sorry, who is he then that maketh me glad, but the same which is made sorry by me? And I wrote this same unto you, lest, when I came, I should have sorrow from them of whom I ought to rejoice;
having confidence in you all, that my joy is the joy of you all. For out of much affliction and anguish of heart I wrote unto you with many tears; not that ye should be grieved, but that ye might know the love which I have more abundantly unto you.
But if any have caused grief, he hath not grieved me, but in part: that I may not overcharge you all. Sufficient to such a man is this punishment, which was inflicted of many. So that contrariwise ye ought rather to forgive him, and comfort him, lest perhaps such a one should be swallowed up with overmuch sorrow.”
2 Corinthians 2:1-7 KJV
just as Paul had written to them for them to exclude the man he had written back again and instructed them to now forgive him who knows exactly how long but I’m imagining his first letter caused a lot of repentance in the church at Corinth and Paul had them forgiven them and the man who transgressed
the sorrow Paul’s letter caused them brought repentance
“Now I rejoice, not that ye were made sorry, but that ye sorrowed to repentance: for ye were made sorry after a godly manner, that ye might receive damage by us in nothing.
For godly sorrow worketh repentance to salvation not to be repented of:
but the sorrow of the world worketh death.”
2 Corinthians 7:9-
if we’re sorry for our sins we are sorry on a godly manner , and this sorrow leads to repentance Paul’s letter sighed heavily on them so much that they had become overwhelmingly sorry and Paul’s second letter shows the “ rod” being layed aside in this matter and he is overwhelming then with love in the second letter even the man who has drawn Paul’s anger before