D&D 5e has moved away from Alignment for the sake of better role playing. Instead of asking "what would a chaotic neutral character do in this situation," it becomes "what would MY character do in this situation." It allows for more character growth and prevents you from putting your character in a box, so to speak. This is why you won't see any spells or abilities that base themselves in alignment.
As others have said, your hit chance does improve with levels via your main attack stat and proficiency bonus. So for example, a level 5 fighter and a level 5 wizard, both with 18 strength, have the same chance to hit with a quarterstaff. However, if a wizard did this, they would no doubt have sacrificed their intelligence score and their spell casting ability. Not only that, the fighter is proficient in more weapons/armor, and thus more versatile in terms of melee combat.
There are some subclasses that offer animal companions. The Beast Master Ranger, any wizard with a familiar, and Pack of the Chain Warlocks all come to mind.
As for the fighter vs paladin question, it comes down to what you want out of a front-liner. Paladins have high saving throws and access to spellcasting by virtue of their class, however, they don't get as many ability score increases or access to powerful feats. Paladins also don't have Second Wind (bonus action heal) or Action Surge (the ability to take another action on the same turn). So, a fighter with 4 attacks could effectively have 8 attacks if their Action Surge is up.
Fighters have some really good subclasses too. Eldritch Knights have some wizard spellcasting thrown in with their fighter abilities, Battle Masters have unique combat maneuvers to control the battlefield, etc.
The question is, which do you find more fun? That's the beauty of 5e...I can't think of a single class I would not have fun playing.
