>Why do some atheists accept the idea of Emotional Intelligence and others do not.<
This appears to be a premise with no basis.
Who is it you're claiming denies that 'emotional intelligence' is a thing and why do you appear to be presuming that it's mostly 'atheists' who hold that view?
>Is it instinct? Or our drive to survive.<
These are synonyms, not opposites.
Genetics create our instinctual drives - which in turn 'present' to our conscious mind as 'emotions'.
We experience 'fear' when we perceive an existential threat - or even an 'ego-threat' which we usually treat as the same thing.
Our reproductive drives present as 'love' for our mates, and 'love' for our offspring. Humans have an exceptionally long maturation period, which necessitates co-parenting, and thus explains why human females can be sexually available even when their not fertile, unlike many other species.
Our instinctual morality is based on the 'tribe'. That which contributes to my tribe's survival odds is 'good', that which detracts from those odds is 'bad'. Which is why 'stealing from your neighbor' is always 'bad', but 'plundering the assets of the enemy' is always 'good', or even 'heroic'. Similarly 'murdering' your neighbor is always 'bad', but 'dispatching the enemy' is always 'good', or even 'heroic'. (note carefully the linguistic tools we use to differentiate the same actions performed upon 'one of us' versus 'not-one of us'.)
N_J