Why do we use pV term independent of U in the equation H = U + pV?
The energy U of a system consisting of a single component and phase is 

$ U = -PV + TS + \mu N$ 

As you rightly point out, U includes the energy required to create the volume by expanding against the surrounding (constant) pressure, namely the -PV term. Removing this contribution gives H. 

Therefore

$ H = U + PV = TS + \mu N$ 

Equating TS with the "heat content" of the system leads to the statement in the textbook (this is my interpretation of this textbook's author's meaning). This kind of algebraic manipulation can lead to all sorts of confusion.

EDIT

Some people recoil (and downvote) when they see an expression such as 

$ U = -PV + TS + \mu N$

as if this were somehow unphysical. Well, it does suggest that there is such a thing as an absolute value of U rather than only relative values, and it is hammered into our heads in school that there are only relative, not absolute, energies. But the point here is that there is an absolute value, relative to an empty system, and you can use N (the number of particles) as your simple scaling constant. The expression above captures the extensive property of U. You could equally write  

$ \Delta U = -P \Delta V + T \Delta S + \mu \Delta N$

and no one will blink. Well, if you define 

$\Delta U = \Delta N \times U_m$

$\Delta S = \Delta N \times S_m$

$\Delta V = \Delta N \times V_m$

with 

$\Delta N = N-0$ 

you obtain an expression equal in spirit to the sacriligeous one above.

Usage of terminology such as "heat content" is potentially confusing, but we can use similar arguments to those used in explaining the PV term's meaning as work to create the system from scratch by increasing the particle number, while keeping intensive properties such as T and P constant, to suggest that the term TS is the heat that must be exchanged with the surroundings during creation of the system (from 0 to N particles). This is more of theoretical than experimental interest, in the same way we can hypothetically charge an object from 0 to a finite charge to compute its energy (relative to no charge).

Note that saying that H represents the heat content is confusing, since the application of the equation is not usually in the spirit I have just outlined, and if you look at 

$ H = U + PV = TS + \mu N$ 

there is an additional term $\mu N$ term which cannot be easily associated with the concept of "heat content". 

So, being generous, I would think the author of the book meant by the statement was, very loosely, that changes in H, that is $\Delta H$, are usually associated with the heat exchanged by a system at constant P (subject also to other restraints but more on that in other questions and answers referring specifically to the meaning of H).