Query min column header while excluding blanks and handling duplicates

Question

I have the following table.

Name	Score A	Score B	Score C
Bob		8	6
Sue	9	12	9
Joe	11	2
Susan	7	9	10
Tim	10	12	4
Ellie	9	8	7

In my actual table there are about 2k rows.

I am trying to get the min score (excluding blanks & handles duplicate scores) for each person into another column using the QUERY formula or ARRAYFORMULA, really to avoid entering a formula for each row.

As I do currently have this

=INDEX($B$1:$D$1,MATCH(MIN(B2:D2),B2:D2,0))

But that involves dragging down through each cell, as I do this on a few sheets that have circa 2k rows, it's very slow when inputting new data.

This should be the end result

Name	Score A	Score B	Score C		Min Score
Bob		8	6		Score C
Sue	9	12	9		Score A
Joe	11	2			Score B
Susan	7	9	10		Score A
Tim	10	12	4		Score C
Ellie	9	8	7		Score C

Answer 1

use:

=INDEX(SORTN(SORT(SPLIT(QUERY(FLATTEN(
 IF(B2:D="",,B1:D1&"×"&B2:D&"×"&ROW(B2:D))), 
 "where Col1 is not null", ), 
 "×"), 3, 1, 2, 1), 9^9, 2, 3, 1),, 1)