AVERAGEIF a value appears in other columns of a row in Google Sheets

I am trying to use AVERAGEIF, or a similar function, to find the average of values in a column if a name appears elsewhere in the row. Normally this is a straightforward use of AVERAGEIF, but the name could appear in a few different columns within the row.

Attached is a toy example.

Here, I want the average score column to be the average of the scores in the rows in which the players appear. For instance, Anna appears in rows 2 and 3, so I want it to calculate the average of the scores in rows 2 and 3.

This is what I want the spreadsheet to show:

In case it matters, the real spreadsheet has more columns that could contain the names, but they shouldn't count outside of the specified columns. so I do need it to restrict to certain columns, not just anywhere in the row.

I got the intended results part to automate the names by doing =sort(UNIQUE(FLATTEN(B2:D4))) (where B2:D4 is the range of the players) in A7 and the appearance count with =countif(B$2:D$4,A7) in B7. The average score is where I am running into issues.

In C7, I tried doing =averageif(B$2:D$4,A7,A$2:A$4) and extending it downward, but instead of what I wanted, I got the following.

It seems like for Anna and Michelle it only took the second score that applies to them instead of the average, but I can't figure out why it just gave me an error for Lois at all.

You can't use Averageif because the dimensions of the criteria range (B2:D4) must match the dimensions of the range of numbers to be averaged (A2:A4). Your formula is just using the first column of B2:D4 as criteria: Lois doesn't appear in the first column so her result is #DIV/0!

In principle you could duplicate A2:A4 three times like this {A2:A4,A2:A4,A2:A4} to match the dimensions of B2:D4, but it doesn't work because this would be an array and Averageifs expects a range.

A simple way to get round this (providing each name can only occur once per row) is to use array multiplication likes this to get the sum and divide by the count:

=ArrayFormula(sum(A$2:A$4*(B$2:D$4=A7)))/countif(B$2:D$4,A7)