Can't COUNTIF an array VLOOKUP without intermediate cells

The attached workbook is simplified from my actual workbook.

On the Wins tab you will see the manual table of number of 1sts, 2nds, 3rds for each player, and next to it the calculated table. (The manual table is wrong, hence the need to move to a calculated table.)

The Wins tab uses an array VLOOKUP(col,table,{c1,c2,c3...}) to get the finished-in-what-place data for each player from the first worksheet and then dumps that into an array of cells. Then the calculated table does a COUNTIF() on that array of cells for each of 1st, 2nd, 3rd.

I thought it would be possible to have a simpler solution without the array of cells, conceptually:
COUNTIF(VLOOKUP(col,table,{c1,c2,c3...})
for each of 1st, etc, except that isn't a valid formula, likely because the array VLOOKUP() output isn't what COUNTIF() is expecting as input. Is there a way to get this to work? (I don't like the intermediate array of cells.)

Or, is there a better way to create the calculated table?

I've researched array formulas via google and have learned enough to know that often a VLOOKUP can be done equally well with something else.

I'm new enough to array formulas to not know what something else looks like.

I recognise that the current structure means one VLOOKUP and three COUNTIFs, and my desired solution without the array of cells means three VLOOKUPs.

...Stu