Mafia 1 has it's own characters unrelated to the other two. Mafia 3 however has a main character and tie in from Mafia 2.
You probably aren't going to like hearing this, but each game has it's own pluses and minuses, and all are in some way worth playing depending on your preferences. There's no one easy answer since they all have different styles which cater to different tastes.
Mafia 1 is very real world oriented in that you must refill your car with fuel, and you can even be pulled over for traffic violations. When stealing cars you must use lock picking. It also has some pretty visceral battles, so you need to be aware of where enemies are and take cover.
Mafia 2 gives you some decent chases, has good story elements, and has some pretty good battles with not too bad DLC, but the story from the protagonist's POV is rather downbeat, whereby he laments over his uneventful life.
Mafia 3 is the most unique in that it's not about the typical Sicily mob, but the Black mob in New Orleans. It easily has the best soundtrack with a lot of rock classics. It brings in some nice new melee moves and gunplay, and great level design, but the whistle lure system can make it too easy. By not using whistle lures, the game becomes far more interesting and less repetitive.
Mafia 3 easily has the best level design and set pieces. It also has some very enjoyable races. In case you're wondering, Mafia 3 is my fave in the series. Just don't use whistle lures much, at the very most, only on the first play through. Then again, if you're OK with a more casual play style, you may not mind being able to easily amass a pile of bodies by just whistling from cover.
You're right that Mafia 3 by default is not well optimized, but I found it's mostly due to heavy processing effects that make the game look very ugly as well. Fortunately a brilliant modder whom goes by the name NyClix, made an excellent mod that makes the game look and run much better. One of the best things about this mod is it doesn't make dark scenes look way too dark like SweetFX does. It's a bit harder to install, but well worth the results.
This is where I accessed and learned how to install the mod. It's a very funny video too. The missing audio in places is just how it was uploaded, the mod in no way affects audio in the game.