Sometimes an english speaker will insist this is a short form of 'I am not going to go'. A prediction or intention rather than an observation. As to how Czech uses the present for the the future, I will watch and learn.
Can't I omit the pronoun and say just "Nejedem"?
Nope, because "nejedem" is not correct. But you can omit the pronoun and use nejedu or nejdu.