Google-sheets – What would I use to write a formula only once and have it work on multiple sheets

Your plans seem sensible to me and I think reasonably straightforward to implement. A formula such as:

=importrange("1NsEAt1biV822J4RnqOmb4MW6GCjrkUzfZjbssnZmkSQ)","Sheet1!A1:F8")

in each of the employee sheets in the Director's cut should import Sheet1!A1:F8 from each individual employee's document (once access has been granted). The long string of characters in the middle of the above formula is the 'key' and is unique to each document.

When all the relevant details are in the Director's copy it should then also be quite easy to process them further for analysis purposes.

In principle, you could do things like

=IMPORTRANGE("1k4o8odd...","Dashboard!S"&row()) 

forming the range string for import based on the row of the current cell.

But I do not actually recommend this. You should not use importrange as an individual cell reference; it's a serious drag on performance. As the name suggests, its purpose is to import the range of all cells you are going to need (maybe the whole sheet you are referring to); put the imported range somewhere (on a separate sheet, perhaps) and work with that data normally. For example, use

=IMPORTRANGE("1k4o8odd...","Dashboard!S1:S")

once and then refer to those imported cells.