Why MonadPlus and not Monad + Monoid?

But couldn’t you rewrite any type constraint of

(MonadPlus m) => ...

as a combination of Monad and Monoid?

No. In the top answer to the question you link, there is already a good explanation about the laws of MonadPlus vs. Monoid. But there are differences even if we ignore the typeclass laws.

Monoid (m a) => ... means that m a has to be a monoid for one particular a chosen by the caller, but MonadPlus m means that m a has to be a monoid for all a. So MonadPlus a is more flexible, and this flexibility is helpful in four situations:

1. If we don’t want to tell the caller what a we intend to use.
   MonadPlus m => ... instead of Monoid (m SecretType) => ...

2. If we want to use multiple different a.
   MonadPlus m => ... instead of (Monoid (m Type1), Monoid (m Type2), ...) => ...

3. If we want to use infinitely many different a.
   MonadPlus m => ... instead of not possible.

4. If we don’t know what a we need.
   MonadPlus m => ... instead of not possible.