Billy could do this all at home and still be saved. Without baptism or Communion. His faith might then lead him to a public confession of some kind and a church and Communion, but those things didn't save him.
There are some that believe that you are saved my the single thought that Christ is real...not me.
Saved and salvation are topics that are not as simple as you might think.
So you might say that baptism does not save you and does not remit your sins and you might say that communion does not save you and does not setup a condition that you can go to heaven.
But what happens if you do not do them?...When Christ was talking about the bread and wine ritual, He said that if you did not do it, that you do not abide in him or He in you and there would be no life in you and so there would be no reason to raise you on the last day....what would it be like to be saved and none of that occur. What would happen to you if none of that occurred. A saved soul with nowhere to go?
The Apostles and everyone since has tried to understand the spiritual side of these rituals, has anyone gotten it right? I believe it comes down to...Luke 6:46 "Why do you call me, 'Lord, Lord,' and do not do what I say?