You can tell them that they are wrong. Blue Line represents Vyse at Max Weight at Sea Level. If you are not at max weight or are at a higher density altitude, then you can improve your rate of climb, or slow your rate of descent by flying at a slower airspeed. The question is how much slower? The Seminole (I am assuming this is what you use at Riddle) unfortunately does not have charts in the POH that give any guidance.
As a general rule, I avoid flight below Vxse. There is usually no useful benefit in going there, and it can obviously be dangerous if you get too slow. At max weight Vyse will always be between the published Vyse and Vxse. At lower weights, you should still probably be able to optimize performance (climb rate) between the speeds published for max weight, especially in a light twin where the weight does not really vary that much. The larger twins that can carry hundreds of pounds of cargo should have a more complete POH that contains charts for this (Vxse and Vyse vs Wt. and D.A.).
Initially when you first loose an engine, it is a good idea to just go for the blue line, get the airplane cleaned up and either trouble shoot or feather the engine. Once that is done, if you really need the performance, it is obviously ok to try to find the airspeed that will give you the best climb rate.
If your instructors would rather fly into terrain than slow down a few knots and clear it on takeoff, that is their business.