In no particular order:
1 - Try start with the bosss favourite coffee made with freshly ground coffee and another with whats left in the hopper from yesterday (Ill bet that currently they leave it there).
2 - Show the boss some espressos made the same way so they can see the lack of crema from the old grinds.
3 - Point out that consistency between baristas is paramount.
As a customer I hate going to a café and getting something made differently depending on whos working the machine.
4 - Give the boss the same taste test with just milk, fresh and reheated.
As I dont do it I cant say for sure but reheated milk must taste worse than fresh.
Actually, go back to number 1 and make the boss their favourite with both stale coffee and reheated milk (if they drink their coffee with milk).
5 - Check the volumetric output; does it overextract? If so, good. The worse a coffee you can make the boss using the current practices the better for your argument. Compare one of those to the best you can do and that should sell your argument.
6 - Find out if the new supplier will do more frequent deliveries of less beans and just argue logically about it regarding the freshness. The quality of the end product keeps customer returning; some from quite some distance.
Does the boss want to be run of the mill or stand out from the crowd?
How much influence will you really have?
Does he really trust you or not?