Does String Theory Really Demand Higher Dimensions?

A real string theorist can correct me if I am wrong, but: it seems that string theory does not require space to be higher dimensional as is usually claimed.

Take for example bosonic string theory of which it is often said can only exist in 26 dimensions.  However, on a technical level, I'm not sure this is correct and perhaps we should stop using this terminology.

What is really critical is that the central charge of your CFT = 26.  For those who don't know, string theory has  conformal invariance and thus has a lot to do with conformal field theories. (CFT)  Conformal field theories have something called a central charge c.  It seems to me, the only requirement for the bosonic string theory to be valid is that you need c = 26.

For example, take a look at David Tong's lecture notes where it is written:

The consistency requirement is merely that the degrees of freedom of the string are described by a CFT with c = 26. Any CFT will do... If you like, the space of CFTs with c = 26 can be thought of as the space of classical solutions of string theory.

Now even more to the point:

We learn that the “critical dimension” of string theory is something of a misnomer: it is really a “critical central charge”. Only for rather special CFTs can this central charge be thought of as a spacetime dimension.

For example, if we wish to describe strings moving in 4d Minkowski space, we can take D = 4 free scalars (one of which will be timelike) together with some other c = 22 CFT. This CFT may have a geometrical interpretation, or it may be something more abstract. The CFT with c = 22 is sometimes called the “internal sector” of the theory. It is what we really mean when we talk about the “extra hidden dimensions of string theory”.

Or consider this quote from String Theory and M-Theory: A Modern Introduction:

One may saturate the central-charge condition in other ways. In critical string theories one chooses D ≤ 26 space-time dimensions, and then adjoins a unitary CFT with c = 26 − D to make up the rest of the required central charge. This CFT need not have a geometric interpretation. Nevertheless, it gives a consistent string theory 

Okay, so assuming I am reading this correctly, string theory doesn't demand there are hgher dimensions, only that the central charge takes on a special value.  The theory can be lower dimensional provided the correct CFT is being used.

If this is true I think we need to stop saying string theory requires extra dimensions and start saying string theory demands special field theories that are highly constrained by the number of dimensions.