Welcome!
Too bad you didn't provide a link to the sample test data - it took a long time to create. I assumed that your table starts with column A and, accordingly, the "Post-Game ELO" columns are in I:L. If this is not the case, then you will need to change the formula, which we will construct together now step by step.
Let's put the name Dan in cell A6 and start building a formula for this player in cell E6.
First of all, let's make sure that the player for whom we are looking for a result is already in the table. If this is a beginner (like Donny and Ivanka in your example), then instead of a number, we indicate x.
To do this, use COUNTIF() to count how many times this name occurs in columns A:D above the current row.
=IF(COUNTIF($A$1:$D5,A6), <a number that we will find a little later>, "x")
Pay attention to the addressing of the search range - the beginning of the range is the absolute cell $A$1, and the end of the range is indicated by the absolute column $D and the relative row 5 - this is one row above the cell with our formula. Due to such addressing, we will be able to drag the finished formula across columns and rows without distorting the range.
Now let's find Dan's most recent result - find the last occurrence of Dan's name in the table, and when we know the desired cell in the range $A$1:$D5, take the value from the corresponding cell in the range $I$1:$L5.
The main problem with your data layout is that the built-in search functions work on one column (row) and you have four.
So let's go for the trick, convert both of these ranges into one column using the FLATTEN() function.
...FLATTEN($I$1:$L5)...
For the range of names, we will make one more transformation - we will replace all names except Dan with 0. In other words, we will search in an array of the form {0;0;0;0;0;0;0;0;Dan;0;0;0;0;Dan;0;0;0;Dan;0;0}
...FLATTEN(IF($A$1:$D5=A6,A6,0)...
We have all the necessary data - we will use the INDEX(MATCH()) construction to get the result. There is a lot of information on the Internet about this method of searching for values, I will not repeat those texts (and videos).
=ArrayFormula(IF(COUNTIF($A$1:$D5,A6),INDEX(FLATTEN($I$1:$L5),MATCH(A6,FLATTEN(IF($A$1:$D5=A6,A6,0)),1)),"x"))
Expand the formula along columns E:H and down to the required number of rows