How to cross reference data in multiple columns then add together data in a specific column?

I have a table full of cases, who they belong too, and how long it takes to get it assigned basically. I am trying to find a formula to check a column for a certain word then follow that row to another column and find the number notated. Then to add all numbers it found together to give me an average. My goal is to find an average number it takes for certain cases to get assigned.

I have been using countif for most of what I need as it only pulls from one column then tells me to the total of times something appears. The cross-referencing is out of my wheelhouse. I've seen recommendations xlookup/index/match/vlookup but not sure which is best or if they will meet the need.