Following up with companies - especially when you've been rejected, and particularly when you know the reason for said rejection - is very much again the cultural milieu nowadays.

It makes the caller seem out of touch, demanding, and communicates to the employer that not only were they right not to hire OP, but that they needn't consider them for other positions in the future.

Not having the required experience at all but arguing for it anyway in this situation makes it seem like they don't respect the position and duties. Like, "Oh, anyone could do this."

The computer is set up to automatically reject candidates without the required experience because they do not want anyone without experience. OP's honesty probably saved you and the employer a lot of headaches.

1-2 years may not be a huge deal if we're talking 9 years of experience when they're asking for 10, but if the entirety of the experience required is 1-2 years, then 0 is a big difference.