Please note the ‘if’. I’m not one of those people who thinks they could do a better job than Wizards of the Coast. The main reason I’m bothering to think about this is that I’m considering writing a game once I get my current writing tasks out of the way.
‘Contemporary fantasy’ here means ‘contemporary English-speaking fantasy’. I know very little about, for example, anime.
Thief-like characters would be ninjas (or possibly dancers – do little girls still like ballet dancers?), or sneaky types of animals such as cats and mice.
Pirates and witches would be ‘core’ class/options.
Angels and demons, vampires, werewolves, dragons, unicorns and fairies would all be ‘core’ options for those games where there are non-human player characters.
Talking animals would be much more prominent- probably based on stereotypes about the animals rather than their actual nature. For example, talking cats would be smarter than dogs.
The big ‘loser’ would be Tolkien-derived classes and races. If there’s a dwarf or halfling-like race/species at all, it would be gnomes.
The setting wouldn’t necessarily be a self-contained world. Perhaps there’s a fantasy world hidden in the real world (as in Buffy, the Harry Potter series, or the World of Darkness RPGs).
The niche occupied by barbarians, half-orcs etc. would be taken by cavemen/women (or talking animals such as bears).
Luck and Courage would be important stats.
A clear, detailed taxonomy, as in D&D, where (eg) ‘orc’, ‘goblin’ and ‘hobgoblin’ each refer to exactly one kind of monster, and each monster has exactly one name, probably wouldn’t exist.
Zombies (the zombies of Night of the Living Dead rather than those of Haitian folklore) would occupy the ‘weak monster that you can only kill’ niche occupied by orcs.
Characters who heal, characters who smite the undead, and characters who are holy might all exist. But the combination of them all in the ‘cleric’ wouldn’t.
‘Racial classes’ would be more likely than D&D’s usual practice of having race and class, with class the much more important factor.
Finally, covers would be based on the way most fantasy novels seem to look, as discussed here.