The reason that Tesla was not
The reason that Tesla was not invited was nominally because it is not a union shop.
Another concern for the administration is that Tesla is seen a symbol of liberal arrogance and luxury.
Sure the Model 3 is an affordable EV but it competes against BMWs, Alfas and Porsche. Not Chevys, Fords and Toyotas. Cars seen as affordable vehicles for the working man.
Tesla could have shown a Model 3 or Y and you KNOW Fox would have commented that Biden was promoting subsidies for people buying $100+ cars and referencing a Model S or X price.
They just do not need that noise whenntrying to advance an agenda that WILL help Tesla while really helping the planet.