SCPM: Some say you have to finish Contra 3 or Probotector on Hard, but I did not finish either one on that difficulty. You don't need to finish any of the 'Bonus' games either. There doesn't seem to be any pattern for what works, other than beating the boss and sitting through the credits again.
I understand what you're saying and I didn't finish all the games on hard, but I did finish every game. I also finished both Super Probotector Alien Rebels and Probotector with all the endings, and I watched all the endings with credits. I always do this with every game I finish with respect to the people behind the games.
I believe I read on Steam that someone posted that you have to finish all the games once to get the Run N' Gun Specialist achievement, and finish them all again to get the True Contra Warrior achievement.
What I'm going to do is remove the game and all my saved games. Reinstall the game and play each game and save the game before each end. Maybe then the achievement will finally pop.