As for Padme, sure, she could probably have arranged for it
If we accept book canon, she did. She sent her handmaiden to free slaves and look for Shmi but couldn't find her. Shmi also got involved with some liberation movements and her old home had some evidence that she was likely already free. So while no one could find her, they knew she probably wasn't a slave which was at least "good enough".
As for Anakin, I'm confident that you're completely right that the Jedi were interested in tackling slavery at a deeper level. I'm sure he believed that if he put in the work and helped the Jedi that they would be able to eliminate slavery.
I think Anakin dreamed of the day he could return to Tatooine not just to free his mother, but to free ALL of Tatooine. He was going off to train and become a big hero, so he'd want to return as a big hero.