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Yes, but it surely does not exclude their membership in an organization, does it?

No, not at all! In fact, Christ urges us to support one another, and to be together. "Where two or more are gathered in My name"...

We have, however, other passages that urge doctrinal unity in the same group Jesus is speaking to.

Doctrine as Jesus taught, yes.

That would make them collectively, the Church.

Sure. But you're missing the bigger picture, the point:

The Holy Spirit is given from God to individuals who may be part of the universal (catholic) church. It is NOT given from God to the church then to individuals.

We worship collectively - even in twos or threes - with the Holy Spirit present in each of us, and do not need the church to provide the Holy Spirit.

God gives us His grace and His Holy Spirit directly, not through the church. The Holy Spirit is present in the church because believers are there, and they receive the Holy Spirit directly from God; if there were no believers present, then the church would be absent of the Holy Spirit.

God gives us His grace and His Holy Spirit directly, not through the church

The scripture shows that the Holy Spirit was givien initally to the apostles and then they were told to teach and baptize others. The distinction was there from the very beginning:

And you shall give testimony, because you are with me from the beginning. (John 15:27)
21 He said therefore to them again: Peace be to you. As the Father hath sent me, I also send you. 22 When he had said this, he breathed on them; and he said to them: Receive ye the Holy Ghost. 23 Whose sins you shall forgive, they are forgiven them; and whose sins you shall retain, they are retained. (John 20)
24 As for you, let that which you have heard from the beginning, abide in you. If that abide in you, which you have heard from the beginning, you also shall abide in the Son, and in the Father. 25 And this is the promise which he hath promised us, life everlasting. 26 These things have I written to you, concerning them that seduce you. 27 And as for you, let the unction, which you have received from him, abide in you. And you have no need that any man teach you; but as his unction teacheth you of all things, and is truth, and is no lie. And as it hath taught you, abide in him (1 John 2)

Here, the charism of teaching, the grace of confession, and the grace of the holy Orders are expressly given the nascent Church and through her to the rest of the believers.