You need to select ALL columns that you wish to sort, i.e. Columns A:I. Then sort by the Total column Smallest to Largest.
One issue you will have is with the 0's in the Total column being sorted as the lowest values, so you'll need to make the Total formulas equal to a blank "" if no scores have been added to that row.
Easiest way to do this is to leave the score cells blank rather than add a zero, then wrap your Total formula in an IFERROR function to equal "".
So...
=IFERROR(SUM(SMALL(D2:H2,{1,2,3,4})),"")
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