> You will pick up enough of multiplication tables through doing maths
You will not do maths casually until you have memorized enough multiplication to make it not torture. You will not pick up multiplication from using a calculator any more than you will pick up programming from using a computer.
> native speakers of a language will conjugate correctly without memorising
They do not. They have memorized, through massive, constant, and forced practice, and now they conjugate correctly. The alternative of consulting a computer every time they need to speak is not a realistic one.
> You will not pick up multiplication from using a calculator
Sure you will, at least assuming we're still talking about memorizing multiplication tables here and not how to do long division or the like. I don't think algebra or even basic calculus has any convincing need to involve rote memorization.
I've ended up unintentionally memorizing many things due to frequently needing to consult various lookup tables.
> conjugation
Competent ones will. Wrong conjugations usually "sound" wrong to me even when I haven't seen them before and that's in English of all things.