When people say church do they mean the building (“we’re renovating the church”) or the gathering (“we’re late for church”) or something else?

The Greek word sunagógé or synagogue is from sun (together) and agó (bring). Literally “bring together”. It covers both the building and the gathering: “an assembly, congregation, synagogue, either the place or the people gathered together in the place.” – Bible Hub

The Greek word for church, however, is ekklésia. Bible Hub breaks it down:

1577 ekklēsía(from 1537 /ek, “out from and to” and 2564 /kaléō, “to call”) – properly, people called out from the world and to God, the outcome being the Church (the mystical body of Christ) – i.e. the universal (total) body of believers whom God calls out from the world and into His eternal kingdom.

For hundreds of years, the Greeks had been using this word as follows:

Ecclesia, Greek Ekklēsia, (“gathering of those summoned”), in ancient Greece, assembly of citizens in a city-state. Its roots lay in the Homeric agora, the meeting of the people. The Athenian Ecclesia, for which exists the most detailed record, was already functioning in Draco’s day (c. 621 BC). – Britannica

If church was primarily a building and/or gathering, why didn’t they call it synagogue (“bring-together”)?

Why did they choose a name that for hundreds of years had been used to describe citizens being called out of regular society to govern?

Could it be that “church” means the people who are called out of the world to reign with God in his kingdom?

• You have made them to be a kingdom and priests to serve our God, and they will reign on the earth. – Rev 5:10
• If we endure, we will also reign with him – 2 Tim 2:12
• Now if we are children, then we are heirs—heirs of God and co-heirs with Christ – Ro 8:17
• Do you not know that the Lord’s people will judge the world? – 1 Co 6:2

Then maybe we gather partly because this is such a mind-blowing calling that left to ourselves we could think we’re crazy. But it’s more than that. Here is the only New Testament verse that tells us to gather. It tells us why we gather:

Let us hold unswervingly to the hope we profess, for he who promised is faithful. And let us consider how we may spur one another on toward love and good deeds, not giving up meeting together, as some are in the habit of doing, but encouraging one another—and all the more as you see the Day approaching. – Heb 10:23-25