"To boldly go where no man has gone before"
"We are determined to completely and utterly eradicate the disease."
"She wants to gradually get rid of her teddy bears."
"Writers should learn to not split infinitives."
It's also not always possible to eliminate the split infinitive, or the infinitive, without changing the direct or implied meaning of the sentence. "She wants to get rid of her teddy bears gradually" works fairly well.
"Boldly going where no man has gone before" is close, but doesn't capture the same challenge, summons, call to action as the original. Nor does it sound as pleasing.
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!
She wants to get rid of her teddy bears gradually.