Formula to get a unique value list and sort one column by numbers in another column

I have a sheet that has a couple of columns that contain some names, and counts associated with those names.

Column A Column B
Person Count
A 1
B 3
A 4
B 5
C 2
D 3
D 1


I'd like to create a formula that I can copy down a third column that will produce a list of unique values from the person column, sorted by the sum of the counts for that person from the counts column.

I.e. the output from this formula would be a column looking like this
Column C
sorted unique list
B
A
D
C


(where the sort order comes from the fact that the counts for B in the original two tables sums to 8, A to 5, D to 4, and C to 2).

It seems like I should be able to create some array formula to do this, but I'm getting stuck when I try.

Can anyone help?

Thanks!

Without any helper columns. In C2 copied down:

=IFERROR(INDEX($A$2:$A$32,MATCH(LARGE(INDEX((COUNTIF($C$1:C1,$A$2:$A$32)=0)*(ROW($A$2:$A$32)/10^10+SUMIF($A$2:$A$32,$A$2:$A$32,$B$2:$B$32)),0),1),INDEX(ROW($A$2:$A$32)/10^10+SUMIF($A$2:$A$32,$A$2:$A$32,$B$2:$B$32),0),0)),"")

Please see attached.