Individual Christian participation today in assembly comprises mostly singing. This was not originally the case. Instead individual expression of faith was “in remembrance” as portrayed in The Lord’s Supper. If it is “a remembrance” then we can conclude this is not an actual sacrifice. Instead, just as the O.T. sacrifices never took away sins (Heb.10.4) so we ‘testify’ that Christ’s sacrifice is our hope. Taking part in the LS then is like a ‘confession’ since we “proclaim Christ’s death until He returns” (1Cor. 11.26). Though this observance can be a witness to non-Christians who observe, primarily this ‘individualized stance’ when we participate functions as an encouragement to other Christians.
Here is some history of the practice from the early years after it was instituted. This seems to continue the original practice: “they devoted themselves to the apostles teaching, the breaking of bread (LS), fellowship, and prayer” (Acts 2.42).