What's wrong with the calc you have? If the Wanted is giving you what you want, what needs to change?

Could use AVERAGEIF but would need to include year in Quarter calculation.

=YEAR(A2) & ROUNDUP(MONTH(A2)/3,0)

Then use that in AVERAGEIF.

=AVERAGEIF($B$2:$B$34,"=" & B2,$C$2:$C$34)

If you want to extract year/quarter from date value for use in formula without having to calculate in a column, sorry, don't know that. Every example I find shows using helper column.