How to add or remove names within one sheet and automatically update others

This won't work at all. A VBA solution is the only possible way to have what you want.

Having had to keep records, I don't like solutions that delete records. It is better to identify their status rather than getting rid of the records. You could do this by going back to the original workbook, add a column for status on each worksheet then you can use the filters to manipulate your records.

I have done this with your original workbook by adding a Status column and marked a few records with an x (just for convenience as you can have anything that you like) to signify a non-active or if you will "deleted" record.