If you replace your numerical values with algebraic symbols, you may see the problem with your analysis -- its use of a zero allows 1 to equal 2, as shown in this classic example:

http://www.jimloy.com/algebra/two.htm

So multiplying by zero isn't always a legitimate operation. That's why the multiplicative identity is one, not zero.

Multiplying both sides by zero is always legitimate (if rarely useful). In the example linked, the error is in dividing by zero. More specifically, in dividing by (a-x). Formally speaking, this operation is only defined when (a-x)!=0. Mutipling by zero is defined for all real numbers.

The parallel I was making is that a numerical term invalidated an equation, rendered it meaningless, not that the circumstances were identical.

I think you're having trouble with the definition of "preserve". If you begin with a false statement, there isn't actually a truth value available to preserve.

Multiplying both sides by zero yields an equation which is vacuously true, and is not useful for solving the problems algebraists set out to solve.

True, but it is still valid algebra. There are other examples where one might turn a false statement into a true statement without doing something as pointless as multipling by zero. A common example is squaring an equation. Consider a system of equation in which it can be shown that x=3. Within this system, the statement x=-3 is false, however the statement x^2=9 is true.

Again, this specific example is contrived, but this does come up annoyingly often.

Sorry, s/vacuously true/tautological/g