Is "贪得无厌" the best choice for this sentence?
贪得无厌 -and/or- 贪得无餍 can be found in almost any dictionary.
ABC defines it as:

be insatiably greedy

Oxford defines it as:

have an insatiable greed

So we can kind of see where Google Translate's insatiable came from.
MDBG has a rather unique translation too that you can consider:

avaricious and insatiable (idiom); greedy and never satisfied

Wiktionary

(idiomatic) greedy, avaricious

I'm not sure where your Chinese excerpt came from but it's not written particularly well to begin with.