"The true Church must be continuous from the apostolic age. There is no room in Christianity for a "gap" or interruption of thousands of years during which true Christianity ceased to exist and then was revived in the form of another movement"
That is definitely not true.
Jesus said in Matt 13 that there would be wheat Christians and when the sower (Jesus and the apostles) died off then weeds would overrun them. After that the differences would not be recognizable. Then in the time of the harvest, the last days the weeds would be discernable from the wheat.
That is what happened. When Russell and the later ones moved toward the truth, the wheat became discernable from the false Christian weeds.