It's not exactly shown since only two or three ever fire at one time (par for the course Trekwise, I know).
One could well argue that any single cannon can channel the total output of the ship for destructive purposes, and multiple cannon exist only because of the need to cover multiple angles, to cool down individual emitters between shots, and for redundancy. After all, just as you say, firing from just one or two emitters
is a prominent Trek feature.
And yet Silent Enemy specifically tells us the ship was DESIGNED to house three of those cannons.
True. But the ship still had thirteen gunports from the very start. Design intent may well have been to carry three phase cannon, five plasma cannon and five railguns, but experience showed that only the first one was a worthwhile weapon - experience only gained through the exploits of NX-01 herself.
Except later in The Expanse" we have Reed saying that a shot from the Klingons has knocked out "both forward phase cannons," reducing the number back to three (unless the other 11 cannons are all mounted aft

).
FWIW, the gunports are grouped so that there are four ahead of the bridge above and four below the saucer, then two in each of the booms (although one in each was originally intended to be this "plasma recharger" thing), then one in the rear pod. To satisfy semantics, we could argue that a "cannon" actually refers to a pair. But then again, two of the dorsal forward ports were never shown firing anything much - except perhaps plasma bolts in "Broken Bow". So "both forward cannon" may well refer to the bowmost gunports, those just behind the deflector dish...
That and "Shockwave Part II" where the beams are firing from the lower cannons yet striking targets ABOVE the ship. It's similar to the error in "Cause and Effect" where the Bozeman rams the Enterprise' nacelle at such an angle that it couldn't have actually TOUCHED the nacelle without hitting the saucer first.
Granted - but that's a somewhat different category of VFX error, similar to scaling problems or lighting issues and IMHO dissimilar to actual "misuse of a feature".
But again, we never saw "thirteen phase cannons" firing on a target. We saw TWO phase cannons at any given time, and their location shifted as VFX artists apparently forgot where they were supposed to be located (as they have on various occasions with various starships over the years). This is exactly like the energy beam from the "phaser bank" in Encounter at Farpoint.
In the backstage sense, it's very different - the VFX people knew perfectly well where they were allowed to have beams, and they used all those options - and, remarkably, never really used anything that wasn't an option (unless we choose to believe in the never-established "turbocharger caps").
The end result may be similar, of course... Except that it isn't. Drexler designed the gunports to house guns (or other devices as needed - I believe Echo One deploys from a gunport, but long before there's a gun there). They did. Probert designed the phaser strips to be phaser strips. Yet beams emerged from the (admittedly noncanon) Yacht and the (established) torpedo tube. So these end results aren't really the same.
Seems like three phase cannons, four spatial torpedo tubes, two photon torpedo tubes and an unknown number of plasma cannons altogether. Not much more than that.
Nope -
sounds like that. Seems something else again.
Did the E-D have one phaser, or two, or eleven? Dialogue never referred to any other number of phasers besides one, even though this one could be in plural at times. Yet we saw eleven emitters in action (if we accept symmetry as proof for those that didn't fire), eleven built on the model, eleven intended.
Dialogue doesn't really contradict the existence of eleven strips on the NCC-1701-D, or thirteen cannon on the NX-01. It may suggest other things if taken alone, but together with visuals it still supports the eleven/thirteen interpretation. It just doesn't make much sense to construe a view of a fictional universe where the reality shifts with the viewpoint (or when viewpoint shifts to hearpoint)...
"The Aft Cannon is online, but just barely."
Actually, it's more like "The aft cannon's online", which may be read as "The aft cannons online" or "The aft cannon (plural) online".
Is it too much of a stretch for the forward cannons to be used out of both the dorsal and ventral tubes? As in the cannon pops out of one or the other.
The ones just ahead of the bridge are quite a bit farther forward than the corresponding ventral ones between the torpedo tubes. However, the ones next to the corners of the deflector dish do seem symmetrically mounted, in a very thin part of the saucer, and could well be single cannon firing above or below as needed.
The ports on the booms are also on the same vertical axis, and could be the famed "aft" cannon, again single cannon per boom. However, the dorsal gunports on the booms don't ever really open for firing - instead, the beams are drawn as emanating from the "turbochargers" a bit farther forward. So we still have to assume two separate emitters per boom, alas.
Minimum number of emitters would be nine, if we accept four shafts that allow both dorsal and ventral firing, and consider symmetry. Possibly eight if we ignore symmetry and say that only one of the booms has a ventral cannon.
Timo Saloniemi