Well, here's the thing:
Is the fighter a general warrior with access to a wide variety of weapons and armor, or is he a weapon master that focuses on one set of equipment?
Is the wizard a general spellcaster with access to a wide variety of spells and incantation, or is she a focused spellcaster who specializes in one school of magic?
Is the thief a general rogue with access to a breadth of skills and resources, or is she a focused delver who specializes in stealth or intimidation or traps?
The cleric should be of a similar scope.
Are they a general divine prayer-maker with access to a variety of different divine powers, or are they a specific type of divine spellcaster who specializes in melee-mashing and healing and undead turning?
What level of focus should a class assume?
It's a specific design choice that'll have some pretty deep ramifications.
Forex, let's say all these classes ware generalists who have an option to slightly specialize. Your fighter can be slightly better with swords than with axes if he wants, but he's better with both than anyone else is. Your wizard can be slightly better at illusion than abjuration, but she's still better with both than anyone else is. Your rogue might be better at sneaking than at lock-picking, but she's still better at both than anyone else is. Your cleric might be better at healing than at calling down divine fire, but he's still great at both.
This means that it might be easier to "take a level of fighter" and have it mean different things for different characters. This means that you don't need a plethora of classes, just a few unique specializtions (assassins are generalist rogues who swap out backstabbing several times in combat for a one-time surprise-round death attack; illusionists are generalist spellcasters who swap out being good at, say, necromancy, for being better at illusion; paladins are clerics who swap out spellcasting for smiting power; warlocks are wizards who use at-will spells rather than daily spells; etc.). It makes the puzzle pieces either to futz with.
It does lead to a weaker archetype. If any bow-wielding dude in light armor could be a fighter, a ranger, a thief, or even a wizard (think: arcane archer), or even a cleric (think: of Corellon), it's a little less iconic than "THIS IS THE RANGER."
But, personally, I think it's a bit better. It's easier for DMs to customize class features than to write entirely new classes ("It's exactly like the cleric but XYZ is replaced with ABC!"), and it's easier to fit a class into a broad range of campaigns and styles if it has a broad base of abilities (okay, in this Vikings-inspired campaign, a lot of fighters use the Barbarian options for rage and such; you found a magic mace, I guess the cleric can use it, even if she prefers the sickle!).
The alternative is to design something unique for each niche, and I think you can go that way to a certain extent, but at its extreme it results in hyper-specializtion, which is a problem for customizability. If clerics are only heavily armored mace-wielding undead-turners who heal, then I need a brand new class to represent my temple virgin priest of the goddess of lust and temptation, or my sailing priestess of the god of storms and the sea. That's a lot of work for a relatively minor change.
There's problems if you go too far in the "generic" direction, too, but I think a 5e that says "Here's a general cleric, and HERE is an archetypal mace-wielding turn-undead healbot cleric, that fits nicely within the umbrella of "cleric"" would be just fine.
...of course, too far in the generic direction and you have a functionally classless game where you just mix and match different abilities to create the character you want. Which isn't exactly D&D. But I don't think you necessarily need to go that far.
