Audio Edit/ACW is a very good tool however there are some quirks.

OP:s comment on things moving around among already edited bars is likely to be a lack of understanding of the relation between User and Auto generated bar lines.

Working with alignment of non consistent tempo is a fairly quick operation.
Start out like Henry C does with some bar lines and have it autogenerate the remainder of the song.
What Henry doesn't do is to adjust when it gradually goes out of sync. So it just question of checking e.g. every 8 bars and move to the appropriate position and the bar lines to left will adjust accordingly. Continue and the song should be ok in a couple of minutes if it's a fairly consistent recording. Will take a bit more time if it's very sloppy time wise.

On the latest version