You wrote:
“There was no organization, and no institution.”
Well, I typed out a long reply with 9 separate points and then the internet ate it. Shame on me for not saving it in Word first!
I’ll reduce everything to this:
There’s no logical way around the following: the sharing and distribution of goods and monies from a common purse or fund, the designation of leaders and levels of leadership, the clear lines between leaders and others, Jesus’ use of parables for outsiders but the straight story for insiders (Apostles and Disciples), the designated tasks and expansion of levels of leadership and tasks (e.g. deacons serving), discussion of issues in committe or conference and issuing of new rules for followers (Acts 15), the power to oust heretics or public sinners from the assembly, the very idea that the assembly has expections about your behavior and you can be ousted for not measuring up, organized efforts (like EVERYONE praying for Peter when he was arrested, the establishment of house churches and finding of helpers, the fact that St. Paul could leave his property with someone and recall it when he needed it, St. Paul keeping in touch and teaching by letters, the use of specialized terms, the use of specialized rites of initiation (baptism) - all of these show there was an organization and an institution and it was and is the Church.
"...all of these show there was an organization and an institution and it was and is the Church."
But, no, you may wish to "reduce everything" to something that Rome has taught, but it is clearly not definitive. In I Cor., Paul is arguing that the Grecian penchant for polished speakers had clouded their recognition of good, solid doctrine. Apollos may have been better on his feet, and smoother with his tongue, but Paul had been taught directly by Jesus. How could there have been a need to explain if everyone knew that there was just one central authority?
Paul's appeal to the Corinthians is to the solidarity of the Gospel itself, not his right to control nor to Apollos' eloquence or superior wisdom. Both just watered/planted, God caused the growth. Paul even notes that some groups did not recognize his apostleship but that did not stop him. Titles and roles meant little to him and he would publicly spank Peter(Gal. 2).
And, this kind of wrestling to see the truth, my FRiend was done in all the ekklasiai (plural). Read the whole story. There was not a neat and clean little "headquarters". There were multiple "gatherings/assemblies", not a central organization.