She points out (that) oxygen levels were not higher than they are today.
No: in your example, the subordinator "that" is optional. With declarative content clauses (that clauses), it is sometimes obligatory, sometimes optional and sometimes inadmissible:
(1) That I need help is clear (obligatory)
(2) I left before that he arrived (inadmissible]
(3) I know (that) it's genuine (optional)
The main place where that is obligatory is where the content clause is subject of the sentence, as in (1).
"That" is inadmissible in a clause that is complement to a preposition like "before", as in (2). Most prepositions exclude "that"; there are only a very few (such as "notwithstanding", "in order" and "provided"), which allow it.
Elsewhere, "that" is in general optional, as in (3). It is more likely to be omitted in informal than formal style, and it is more likely to be omitted after short and common verbs than after longer and less frequent ones. For example, in This will demonstrate that it is genuine, "that" would probably not be omitted.