Okay, I know this is not the most elegant solution (uses OFFSET and is an array formula for a large spreadsheet), but I thought it was a interesting challenge. This is what worked. Put this in cell A1 of Sheet2:
=OFFSET(Sheet1!$A$1,ROW(),SMALL(--(Sheet1!$A$1:$AX$1)*COLUMN(Sheet1!$A$1:$AX$1),COUNTIF(Sheet1!$B$1:$AX$1,FALSE)+COLUMN()+1)-1)
Enter with Ctrl-Shift-Enter, and then drag across and down the sheet.
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