Riiiiiight.
Well, it would. Because if you admit that the Spirit only indwells believers, you would have to admit that 1 Corinthians 2:13-14 supports the idea that the Spirit must move in us before we can decide to come to Christ (which you actually admitted a bit ago, but with the flawed idea that it comes to all people and so we have a choice apart from anything in God's will).
Well, we don't. The natural man without the spirit can't know the things of God. Only the person with the Spirit can, and we only get that once we are a believer. Therefore, God must be a "first mover" in the area of salvation. It doesn't mean we are robots who are forced to come. We still want to come to Christ, but only because the Holy Spirit has worked in us, which did not happen to the natural man who stays in his dead and fallen state.