Using Polish as an example, why in personal nouns like "robotnik>robotnicy" or "włoch>włosi" Second Palatalization takes place in the nominative plural, but in non-personal nouns like "ptak>ptaki" it doesn't?
That's not a common-Slavic, but purely Polish phenomenon, since Polish distinguishes masculine persons' vs. other nouns' declension patterns in the plural (human vs. nonhuman). In other Slavic languages the results of the Second Palatalization are more consistent:
Bulgarian вълк (vǎlk) “wolf” – вълци (vǎltsi) “wolves”.
The existence of Polish forms like ptaki can be explained by forming the Nominative plural by analogy with the Accusative plural that originally had the case suffix -y [ɨ], which didn't trigger the Second Palatalization, and then by the 17th century that -y became -i again, *ky is an impossible letter combination in modern Polish, at least in native words.