Do you truly believe a player that passes the ball into the BC has player control?

If so, you really need to re-read Rule 4, Definitions.

A dribble is not a pass. That's in Rule 4, too. :)

First, you don't have to have PC to have a BC violation. Team control is sufficient.

Second, muff it, lose control, fumble, pass, or just stop dribbling while running toward backcourt and then letting the ball continue into the backcourt with out touching it are all the same thing.

I am just saying someone on the offense would have to touch it after it touches the mid court line or in the backcourt before a BC violation should be called.

You should probably consider all of them a loose ball per the definition in Rule 4. ;)

He didn't say "throw", he said "bat" as part of a dribble.

This is the same as the old illegal dribble question.

A1 has ended his dribble, and then pushes the ball to the floor again. Is it a violation as soon as he pushes it down, or when it hits his hand?

Or the OOB call:
A1 dribbling along the sideline. After he pushes a dribble down, he steps on the line. Is it a violation as soon as he steps on the line, or when he touches the ball again thus estalishing the continuation of his dribble?

On that one, we have a specific "Note" in the rule book that says it is OOB when a dribbler steps OOB whether they're touching the ball or not. We don't have a similar rule regarding the division line. Does the concept apply? By letter of the rule, no. By spirit, probably.