Here's the problem with trying to say "X is a theme or background, not a class" is that if you work hard enough, you get left with very few classes.
Lets say WotC takes all the PHB1 classes (Fighter, Cleric, Wizard, Rogue, Bard, Ranger, Paladin, Druid, Illusionist, Assassin, Monk, Sorcerer, Barbarian, Warlock, Warlord) and decides to try to make some Themes or Backgrounds instead.
Its really easy to make warlock a theme (eldrich powers) and background (pact) so its gone. Sorcerer and Ilusionist could also be replicated with just a Theme and proper spell selection.
Bard could be a background (with music/knowledge talents) applied to a rogue (troubadour), wizard (loremaster) or fighter (skald). Assassin likewise is really just a theme (with lots of killing powers/feats).
A druid could a darn good example of a specialty priest, so lets make that a nature background and a shapechanger theme. A monk could work just as well as a martial arts theme and acrobat background; one could imagine priestly monks (shugenja), rogue-monks (ninja) or warrior monks. Even wizard monks (super sayians!)
A ranger is an archer theme (or two-weapon theme) + a natural background. Barbarian is the same, but with a rager theme. Even a Paladin could be a fighter or cleric with the "smite evil" theme. Warlords are pretty much akin to our warrior-bards above but more shouty and less singy.
That leaves us: Fighter, Cleric, Wizard, Rogue. But wait! What's a rogue but a skill-monkey and sneak attack? You could make them a theme (stabby) and a Background (criminal) and have fighter-rogues and wizard rogues. Even our buddy the cleric could theoretically just be a theme/background if not for the arcane/divine magic split.
Hmmm... Fighter, Cleric, Wizard. Where have I heard that before... Oh yeah, OD&D (1974).
....
Its probably for the best at this point to accept ranger, assassin, and all the other PHB1 classes as being unique special snowflakes, even if they could in theory be themes or backgrounds. What is one person's indespensible class is another's talent tree in X supplement.
You could even call it two classes as Clerics are just using magic with a different flavour from Wizards.
Taken to the extreme, you could turn everything into a set of lego blocks that you assemble as you like.
Look at Mutants and Masterminds, True20, or several others like Blue Rose that basically have done this already.
If you peel the class system apart enough then you would end up in your suggested possibility.
It is one end of an extreme line of choices ranging from totally fixed and rigid class builds (you get what it says you get and no more or less). This was typical of OD&D and 1st ed DnD.
2nd ed DnD and 3e DnD worked to change this trend and reduce rigidity by giving options. PF has also followed in this tradition which has made it very successful.
Core elements of a class are things that are packaged to make it easier for a player to choose. They are ideas that makes it easy for me to talk to a new player and help them decide what they would like to do.
I did this just on Friday as I helped a new player figure out from what games they had played in the past and what kind things they wanted to do.
The player wanted to sneak around, ambush people, and possibly set traps. The player like the idea of being tough and hitting people rather than relying on speed to hurt people. This looked like it could be Rogue, Barbarian, or Ranger.
I further asked if he wanted to run in and go face to face with opponents or attack from the shadows. The player preferred to strike from the shadows but didn't want to be a wimpy halfling. I could then assign Orc or Half-Orc (playing PF) as choices along with Rogue.
I further suggested the customization of the Thug choice from Advanced Player's Guide to accentuate the brutal side of the Rogue that liked to use a Great Axe to do Sneak Attacks.
The Class is Rogue and has the Sneak Attack and access to a variety of skills which the player choose the physical skills and avoided the Wis/Cha skills (Wis 6 & Cha 5). Race is Orc (extra strong, intimidating, and knows how to use Great Axes).
The Theme is Thug. The player could play without the theme and the class would work find but using Thug allows the Player to switch a few class abilities to things that more represent what they want to do in the campaign.
Profit to the GM?
1> I got the player selected on a class with only a few questions and could discuss other options with a few questions.
2> A player that is able to more closely get a character to match their wants is likely to play that character longer and with more personality.
3> A player that is more involved with their character tends to pay more attention to the game and cause less problems.
4> A better sketched out character makes it easier for me to work storylines to meet what the players want to do in the game and give them challenges that match their selections of skills.