Yes you are right. Color neg gets less grainy when exposed more. You may get very good results by setting the camera meter to overexpose 1 or even 2 stops all the time.
The newer films include new technologies such as 2-electron sensitization, dye layering etc. to make smaller grain at higher speed.
The aging of the film also affects the shadow grain greatly. So, those old films may have had considerably less grain when they were fresh.