Using CHILDREN for ranges in VLOOKUP or INDEX MATCH

Hoping someone can help with this as it seems possible... I have a sheet for contacts at individual centers across a state. There are roles assigned to the contacts, so for example there may be a director, a supervisor and a coordinator.

In the sheet, each parent row is a row with general center information and (the part I'm having trouble with) a roll-up of contact information from certain child rows. The rows with individual contact information are added under as children. I'm attempting to look in the child rows to find the row for the role of, say, director and then return the column on that row containing their name. In the end I can export a report of just the parent rows (to do some further office automation/mail-merging).

My best estimation is that this can be accomplished by either the VLOOKUP or INDEX/MATCH (or maybe COLLECT?). When I attempt to define my ranges using children, the formula will return unparsable.

Additional information: The intent with updates will be to collect form responses into this sheet. Staff will get notified and the update is moved to the correct center/parent row as a child row, and the old row will be deleted.

In my example, my parent row will have some general information, and then I'd like to "roll-up" the Director's First and Last name. The screenshot shows my attempt at each method, here's what I'm using in the parent DIR FNAME column:

=INDEX(CHILDREN(ROLE@row), MATCH("Director", CHILDREN(ROLE@row)), 4)

And here's VLOOKUP in the parent DIR LNAME column:

=VLOOKUP("Director", CHILDREN(ROLE@row:LAST@row), 2, false)

I think my greatest issue is using CHILDREN and @row appropriately to define the range. Any help would be greatly appreciated!