For the stuff inside the split infinitive, it can to tough to know where else to put it.

So, what would we want?

"Boldly to go where ..."

or

"To go boldly where ..."?

To be really clear that 'boldly' modifies 'to go', it's easiest to split the infinitive.

Your

"She wants to gradually get rid of her teddy bears."

is terrific! There's just no good other place to put 'gradually'!

In case you misunderstood me, for grammar for English, really I have no sympathy for what the heck Latin did!