But the c² isn’t contrived; if you leave it out then you get the incorrect answers. We can do experiments where we convert mass into energy, or energy into mass, and we can measure exactly how much energy or mass is created.
As I said above, if you set c=1 then the constant _can be removed from the equation_ and thus the algebra is simplified, but don’t lose sight of the fact that it is still there: it has just moved into the definition of the units instead. A good example is the Plank unit system, but there are others which make similar simplifications.
Let’s consider a concrete example. The largest amount of energy the average person is likely to deal with on a day–to–day basis is stored in the gas tank of their car. Each kg of gas has about 46MJ of energy, so a full tank is easily several gigajoules. My car has a 16 gallon tank, and each gallon has a mass of 2.86kg, so that is 2.1GJ (2.1×10⁹ Joules). This amount of energy is large, but if we converted it all into mass then the mass would be very small: just 2.34×10⁻⁵ grams. A piece of matter weighing that much would be invisible.
In Plank units, by sheer coincidence, this is very close to 1 Plank Energy; it comes out to just 1.076 Plank Energy. In this unit system, E=m, so this is 1.076 mass as well. But the Plank Mass is also very small! When we convert it to an SI unit, the conversion factor requires us to divide by a big number. We still get 2.34×10⁻⁵ grams grams, because that big number in the conversion factor comes from the original value for the speed of light. Another way to look at is is that my gas tank holds 1.95 billion Plank Mass of gas. This mass unit is too small to be convenient.
This is why we say that a small amount of mass can be converted into a large amount of energy: whether you look at the c² in Einstein’s equation or the large conversion factor from Plank units to SI units, the conversion factor between energy and mass is very large.