I think the answer might be
Need to reinforce your brain?
To stress means to give particular emphasis or importance to something. Stressing an idea reinforces it in your mind.
Need help going with the flow?
Flow stress is the instantaneous value of stress required to continue plasticly deforming a material.
Then I am exceptional for the job!
Occupational Stress is stress related to one's job. Also notice the exclamation mark and the word exceptional which provide a stress to the sentiment.
I am not a coward, but my family is.
Not sure about this one but I think it's referring to a hypothesized family of innate emotions (see here), including acute stress reaction, anxiety, fright and fear.
I have an extravagant sense of humour,
One definition of humour is just a mood or state of mind and a stress is a state of mental or emotional strain or tension, so, in that way, stress is a sense of humour. The word extravagant here again adds stress to the notion.
I can be within precision, masters, and drastic-ness,
This is to do with the stress on syllables in a word i.e, precision, masters, drastic-ness. Acquiring precision or mastery could also involve quite a lot of stress as would a drastic activity.
Most people see me and think, “It’s what is on the inside that counts!”
In mechanics, stress is a physical quantity that expresses the internal forces that neighbouring particles of a continuous material exert on each other.
Yet leave out an important part of me!
Perhaps the external metaphorical stress (forces) experienced?
Well, I am not gonna exaggerate,
Stressing and exaggerating are closely related, but very different, ideas (almost antonyms).
But people believe I am an important part of a holiday, yet I am not.
When we think of the build-up to a holiday, such as Christmas, that could involve a lot of stress in preparation. In the end, holidays are supposed to be stress-free.
Still people may have me everyday,
Yes, a lot of people have stress in their everyday life.
Since the beginning of time?
Stress is synonymous with pressure or tension so you could say the Big Bang resulted from an extraordinary amount of stress at a single point.
Notice the stress on the word exaggerate and repetition of an idea which is another stress.