Thank you for the knowledge, including the link. They may be the closest I will get to an answer. (By "mostly" I indeed mean a density greater than 1/2.)

So density might not be a test for the rate of finite polynomials generating primes or not, like dividing zero by zero. Is the answer different for transcendental polynomials, infinite or not, of irrational coefficients?

Here's another: does an asymptotic density of zero for primes mean that p(1)/1+p(2)/2+p(3)/3+p(4)/4+p(5)/5+...+p(N)/N=2/1+3/2+5/3+7/4+11/5+...+p(N)/N is finite, where N is an integer?