Why does negating Lady Assailant of Flames' effect still require banishing cards?
As the only time this card was released was back in 2003, it is using an older wording standard. This effect can be broken up into two pieces:


  Remove 3 cards from the top of your Deck from play


and


  inflict 800 points of damage to your opponent's Life Points.


The key point of this effect is that it uses the word "to," which was the old way to indicate a cost. "[Do X] to [Do Y]" - X is the cost, and Y is the effect. Modern cards use a semi-colon instead of the word "to" to make a cost more explicit. If this card were re-released, its text would likely read like this:


  FLIP: Banish the top 3 cards of your Deck; inflict 800 damage to your opponent.


(These rewordings were done using a standard Konami called "Problem-Solving Card Text": an overhaul of effect wording to help clarify scenarios like this. Specifically, see bullet point 2 under "Action Structure" of this page for the guideline that relates to identifying costs)

When an effect is negated, paying its costs are not (see Activation Cost, paragraph 4), . Since this card does not use wording that implies using this effect is optional (such as "You may remove 3 cards...", see paragraph 3 of that previous link), the effect must be activated and its costs paid unless it is impossible to pay the cost (such as if you had less than 3 cards in your deck). Element Saurus then negates only the effect part.