At its essence, in the modern understanding, a force is an emergent phenomena arising out of the fact that a world (a spacetime filled with your particles) where two particles of opposite charge seem to move towards each other is more probable than a world where they don't.
This sounds silly but it's exactly the root cause in the current understanding and shoehorning in the word "force" in "force-carrying particles" is a stretch and causes this confusion. It's true that there would be no electromagnetic force without the photons. But photons and their likes are not the only way a "force" arises. For example, the Pauli exclusion principle can be seen as a "force" and it arises without photons with just electrons.
This sounds silly but it's exactly the root cause in the current understanding and shoehorning in the word "force" in "force-carrying particles" is a stretch and causes this confusion. It's true that there would be no electromagnetic force without the photons. But photons and their likes are not the only way a "force" arises. For example, the Pauli exclusion principle can be seen as a "force" and it arises without photons with just electrons.