The laws are important though. They ensure that the "statement-like" things we build out of expressions do behave like statements.

For example, it would be weird if we had three statements A B C that wrote to console, and sequencing A with B and then sequencing the result with C resulted in a different program than sequencing B with C and then sequencing A with the result.

Or, for non-IO things: it would be weird if we had three levels of nested lists, and concatenating two times "from the outside" resulted in a different list than concatenating each inner list first, and then the outer list.

On the contrary, the significance has everything to do with the category-theoretical pattern, albeit the reasoning for that is not immediately obvious and has to do with a further category-theoretical tool known as the Kleisli category. A monad with no enforced "outside" structure is known as a free monad, and is basically the equivalent of what most people outside FP call the Command pattern (or rather, a well-behaved version thereof). And "Commands" have everything to do with specifying side effects!

The historical context helps here. Monads came to Haskell to deal with IO. Once there, people realised that a whole host of things are monads under the hood, most of which are pure. The word "effectful", somewhat confusingly, is often used for applicative/monadic code that is in fact pure under the hood. A good example is the State monad which does not mutate state in any way, it's completely pure computation that happens to model state.

In other words, IO Monad Considered Harmful. https://blog.jle.im/entry/io-monad-considered-harmful.html