Having important characters during your first contract would make story missions weird. Like, if Gottardo was there during your first contract, why would you be surprised seeing a guy with a flail during the beach house mission? And everyone thought Joey was dead until "I'll protect you" anyway. Which leaves Pelata/Gray/Biondo, intro soldatos and Luna for your tutorial
You have a point. Yet with Gottardo, I can assume, that he simple see you and find you intresting enough to tag around.
For why are you supprised that he would be an Serpent and so, well he is a werdo, isn't it?
For some super soldiers of the Delucas, I would more image them like Mr. High Five Guy than Gottardo.
Also some things would be easy change/adjusted with a few lines of text.
For who is going to babe-sit in our first mission. Obvious it could be any (elite) soldato or any higher-ups. Most likely it would be not a caporegime, capo bastone, consigliere or the donna, but we aren't the typical sodlier either. Also I could be that Antonio show us the ropes or Luna. I don't think that Wilfred, one from his unit or our donna would do it. Yet it could even be Gracie if she stumble over the MC and decide to hang around.