How can I better model the relationships in this project?
I am still learning myself so you might not agree with everything I say.

In my business we have 'projects' that last for a certain amount of time and have staff assigned to them.

I would give your Employees a 'default role' eg Manager, Technician whatever. Then when they are added onto a project this role will be used by default in a drop-down menu unless you choose otherwise. This role may affect their pay rate for each project also.

If you really wanted to impress them you could make an application in Visual Basic for creating and updating the tables, you can do this for free I believe.