Is it true that non mutually exclusive events can only be independent events?
No. It is not always true. See explanation.
First we have to look at the definitions.
Two events are mutualy exclusive if their product is empty set
Let's consider 2 events with throwing a 6 sided die.
These 2 events are not mutualy exclusive because their product is:
Now let's check if the events are independent:
Example events are not mutualy exclusive and not independent, so it is a counter-example against hypothesis stated in the question.