"Velocity is $\sqrt{Drag}$" is not a meaningful statement. It confuses causality and is simply wrong.
Drag (at fixed drag coefficient) is proportional to Velocity squared. $D=0.5 \rho\, C_D\, V^2\, S$. However, when you change velocity, you must also change your lift coefficient -- which changes the drag coefficient. The parasite term of drag is proportional to $V^2$, but the induced term ends up proportional to $1/V^2$.
At cruse, when thrust is equal to drag, we sometimes call the power associated with thrust the "Thrust power". It is equal to the product of thrust and velocity. Full stop. $P=T\, V$. Since drag is equal to thrust, we can also think of this as the "Drag Power" $P=D\, V$. It can be thought of as the rate of energy dissipated into the air due to drag.
If you want the power associated with flight, this is it. You're done here. However, you also want to see the connection from here through to fuel flow.
The propeller efficiency (say $\eta_P=0.8$) means that the shaft power is larger than the thrust power $P_{shaft}=P/\eta_P$.
The energy contained in the fuel is characterized by the lower heating value ($\mathrm{LHV}$). We think about fuel consumption for a piston engine in terms of brake specific fuel consumption ($\mathrm{BSFC}$).
BSFC is commonly used in the terrible units of lbm/(hp⋅h). I.e. pounds of fuel consumed per shaft horsepower each hour. So, the fuel flow rate looks something like:
$\dot{m}_f= \mathrm{BSFC}\, P_{shaft}$
The $\mathrm{BSFC}$ is related to the $\mathrm{LHV}$ by the cycle efficiency of the engine $\eta_c$. Obviously a combustion engine is not perfectly efficient -- it gives off substantial waste heat to the air (whether air cooled or via the radiator if liquid cooled) as well as waste heat in the exhaust. All that waste heat is lost energy that was originally contained in the fuel. The cycle efficiency of a piston engine is very low -- typically less than $\eta_c < 0.3$.
$BSFC=\frac{1}{\eta_c LHV}$
We can combine these equations together...
$\dot{m}_f= \frac{D\, V}{\eta_P \eta_c LHV}$
Or, another step further...
$\dot{m}_f= \frac{0.5 \rho\, C_D\, V^3\, S}{\eta_P \eta_c LHV}$
If we assume a simple parabolic drag polar $C_D=C_{D,0}+K\,{C_L}^2$, then we get...
$\dot{m}_f= \left(0.5 \rho\, C_{D,0}\, V^3\, S + \frac{K\ W^2}{0.5 \rho\, V\, S}\right) \frac{1}{\eta_P \eta_c LHV}$
This gives a parasite power term that is proportional to $V^3$ and an induced power term that is proportional to $1/V$.
IMPORTANTLY, this is not the whole story. Buried in this are the facts that propeller and cycle efficiency are not constant. Propeller efficiency $\eta_P$ will vary with speed and throttle setting (power). Cycle efficiency $\eta_c$ will vary with throttle setting (power). In particular, most engines are most efficient operating at near maximum power -- and are very inefficient operating near idle.
If you look at a given aircraft's POH and consider the fuel flow values, you will not see perfectly $V^3$ behavior because of many things -- there is also the $1/V$ term -- and the variation of $\eta_P \eta_c$ is complex and nonlinear.
Edit Below
If we divide the fuel flow by $V$, we get the specific range -- a fuel economy measure like MPG for a car.
$SR= \left(0.5 \rho\, C_{D,0}\, V^2\, S + \frac{K\ W^2}{0.5 \rho\, V^2\, S}\right) \frac{1}{\eta_P \eta_c LHV}$
Which will work out in ft/slug or m/kg, but can be thought of (converted to) nmi/lbm or mpg.
You repeatedly assert that observation of POH data leads you to conclude that various behavior is linear. This is an erroneous conclusion. Taylor's series for any smooth function is:
$f(x_0+\Delta x)=f(x_0)+f'(x_0)\Delta x+\frac{f''(x_0)}{2}(\Delta x)^2+\frac{f'''(x_0)}{6}(\Delta x)^3+...$
When $\Delta x$ is small, then $(\Delta x)^2$ and other higher powers will be smaller and smaller such that they can be ignored. This leaves us with a linear function:
$f(x_0+\Delta x)=f(x_0)+f'(x_0)\Delta x+...$
I.e. everything is linear on a small enough scale.
And on a slightly larger scale, you can keep the quadratic terms and drop everything higher...
$f(x_0+\Delta x)=f(x_0)+f'(x_0)\Delta x+\frac{f''(x_0)}{2}(\Delta x)^2+...$
I.e. everything is quadratic on a slightly larger scale.
If you are looking at POH data, the manufacturer will supply data points at a resolution such that a pilot can linearly interpolate between any given points (or possibly just use the nearest point) to get a reasonable answer. As you point out, we know that fuel economy is bad at low speed -- so the POH does not give data there. It is not useful to a pilot.
However, if we are looking to understand the physics of flight, then we prefer to look at a wider range of data. A range sufficient to show the higher order terms and the more complex interactions.
Edit 2 Below:
The claim was made that thrust can not increase faster than V^2. Below is a plot of thrust available and thrust required (drag) for an F-5E at 36,089 ft altitude.

If this is insufficient to show that drag can increase faster than $V^2$, we can overlay some analysis...
Here I take the lowest drag point -- since this line corresponds to L=W, it is the point of best L/D. If the drag polar were of simple parabolic form, induced and parasite drag would be equal there. I then plot lines proportional to $1/V^2$ and $V^2$ that go through the half-minimum-drag point. I also plot their sum.

The gold line does increase slower than parasite drag's $V^2$ -- because of the $1/V^2$ term of induced drag.
We also see that neither simple curve keeps up with the real drag. This is because of the changes in $C_{D,0}$ and $K$ with Mach number and the non-parabolic parts of the drag polar at high $C_L$ (for the low velocity region of the curve approaching stall).
In the real world, these non-ideal behaviors can drown out the idealized trends. In this case, in the transonic region, $C_{D,0}$ more than doubles between M 0.9 and 1.1 and then remains nearly constant supersonically.
For a piston prop, you won't experience this -- and admittedly, it is one of the most drastic changes in the physical world. They called it the sound barrier for a reason. Unfortunately, I don't immediately have my hands on a similar original chart for a piston-prop aircraft.
Note that the textbook approximations for jet aircraft performance would also tell you that thrust available from a jet is constant with Mach number. It clearly is not.