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Sure, God "could" lead the Church without man-made institutions.

Just like

He "could" have saved Jacob and all his children and grandchildren from famine at the time of Joseph, without Joseph and all his appointed Egyptian officers
He "could" led the Hebrews out of Egypt without Moses and Aaron and the elders from the 12 tribes they appointed to assist them
He "could" have defeated Amalek without Joshua and his warriors
He "could" have extirpated the Midianites without Gideon and his 300 chosen men
He "could" have administered justice without human judges
He "could" have instituted worship in the days of Solomon without priests and Levites and a Temple and the whole elaborate administration thereof
He "could" have spread the Good News of the Kingdom without 72 preaching disciples (in the Gospels), and 12 Apostles and their successors (such as Matthias and Timothy and Titus) who received their ordination by being appointed and by the laying on of hands
He "could" have run the early Church without all the folderol of deacons (Acts 6) and Councils (Acts 15)....

But as it actually happens, He chose, and still chooses, to lead His Church through its appointed leader. And they had, and still have, the actual authority to govern: "It seemed good to the Holy Spirit and to us..." (Acts 15:28).

2 Timothy 1:11
I am appointed a preacher, and an apostle, and a teacher of the Gentiles.

Titus 1:5
For this cause left I thee in Crete, that thou shouldest set in order the things that are wanting, and ordain elders in every city, as I had appointed thee

1 Corinthians 12:28
God has appointed in the church

first of all apostles
second prophets
third teachers
then those who perform miracles
those who have gifts of healing
assistants
administrators
and those who speak various kinds of languages.

He could have done without anybody at all; but He chose to work through the human administration He appointed, which has continued through the centuries from then until now, with many different forms, many different roles, many different ways of being organized (like the Vatican), but one Spirit, until He comes again.