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.
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.
Doing maths is not torture if you do not know multiplication tables if you have a calculator.
Native speakers of a language do not memorise conjugations through forced practice, they memorise through hearing them repeatedly from others.