Why we can't have some algorithm to be polynomial if there are generic conditions that make them so?
It's often the case that an NP-complete problem has some parameter $b$ that when fixed to a small constant makes an polynomial algorithm possible, yes.
However in fixing $b$ that problem (at least as of writing when $P =^? NP$) loses its completeness. That means when we want to do a reduction we (for at least some instances) need this parameter to be bigger than $b$.