Why Didn't Palp Just Go Get Anakin and Train Him?

The more obvious answer is Palpatine wanted to ascend through the political ranks of the Senate to Supreme Chancellor, and had not the time to search for Plagueis’ creation: Anakin.

The more insidious reason is Sheev wanted Anakin to learn the Jedi ways, where all the records where, temples, and even Jedi powers so that he could more easily institute Order 606 the Jedi Purge.

 

Actually (and maybe that was already said) I can imagine that if he knew about Anakin (or even was involed in his creation) his plan might have been to make him his apprentice, when he's old enough, so that the Jedi aren't interested in him any more at all (because he is to old). I don't think for a Sith there is no requirement to start training while still being breastfed. When the force made the Jedi discover Anakin early he had to adjust.

 

Qui-Gon found Anakin first.

 

You are right. Sith have no age requirement for training. Ulic Qel Droma was an adult when Freedon Naga called him to be Exar Kun’s apprentice. In Canon Count Dooku was made Darth Sidious’ apprentice at an advanced age.

The Altisian Jedi do not follow the cradle rule for selecting Jedi and training them. They will train full adult apprentices. Yoda and Obi-Wan adopted an Altisian approach by training Luke when he is already an adult.