In the term to the right of the equal sign, the c's cancel out. And m squared divided by m is just m.
So E= c*m^2/m*c simplifies to E = m. That makes no sense.

E is in Joules, a unit of energy, and a Joule is defined as a kilogram meter squared per second squared.

Do the dimensional analysis, substituting the units for the quantities:

E = Mc^2 where capital M is Mass, lc m is meters.
E = M (v)^2 c is just another velocity, v = meters per second
E = M (m/s)^2
E = kg (m/s)(m/s)
E = kg m^2/s^2
Joules = kg meters squared per second squared

Einstein had to multiply mass by a squared velocity or the answer wouldn't come out in energy units. In your alternative formula, the answer comes out in gibberish.

Now an equation coming out in the right units doesn't necessarily mean its correct. But if it doesn't come out in the right units, it is necessarily wrong. For example, the classical vis-viva energy equation E = 1/2 mv^2 yields the correct values and the correct units, in the non-relativistic case.

Einstein didn't just come up with an equation that gave the right units and decided it was right. He derived that equation from other equations that had already been demonstrated to be correct. I used to know that proof when I was a student, but I can't remember it now. Too bad, it was quite simple, and quite elegant.