The kutta condition can be looked at a couple of ways - the way it is often used is to effectively "correct" an inviscid flowfield. However, the reason why the kutta condition holds is because of viscosity. In the (admittedly nonphysical) limit of an infinitely sharp trailing edge, in the absence of the kutta condition, the flow around the trailing edge causes an infinite shear. Of course, for a real airfoil, the trailing edge is not infinitely sharp, but so long as it is sufficiently sharp, an extremely high shear would still develop in this case. Any viscosity at all (even a very small viscosity) will cause large forces to act on the fluid due to this shear, and these forces are what actually cause the flow to come smoothly off both the top and bottom surfaces of the wing at the trailing edge. So, viscosity is the physical reason why the kutta condition holds.
I agree that the kutta condition needs to be employed to model the effects of viscosity however, your explanation about viscosity allowing flow to leave the trailing edge smoothly is incorrect.
As for the statement that flows with viscosity won't leave the trailing edge smoothly? Why do you think that? Flows with viscosity can flow smoothly off the trailing edge just fine. Depending on the details, the boundary layer will typically be turbulent for most aircraft, true, but this is on the scale of millimeters - the overall flow pattern is still smooth and attached.
Any airfoil at a non zero angle of attack (well this isn't 100% true but for most "normal" airfoils it is) will have some degree of separation at the trailing edge. Obviously this becomes more prevalent at higher angles. Take a look at some pictures of google of airfoils at different angles of attack, notice that even around 5 degrees the separation point can be around 80% of the chord. The flow, therefore cannot leave the trailing edge smoothly. If this were true then that would imply that there is a stagnation point at the TE and no separation will be present.
I'm also not sure why you're bringing total pressure into this. It's true that there's energy loss in a boundary layer, but to a pretty good approximation, the flow outside of the boundary layer itself is inviscid and lossless (for aircraft). As a result, the Bernoulli relation can be used along with the air velocity just outside of the boundary layer to determine the pressure just outside the boundary layer (which can be found by running an inviscid simulation of the airfoil, or if you really want to be picky about it, you can replace the airfoil shape with a slightly modified shape that adds in the boundary layer displacement thickness). In boundary layers, the pressure gradient in the normal direction to the surface is minimal (and very frequently assumed to be zero), so knowing the pressure distribution just outside the boundary layer also tells you the pressure distribution inside the boundary layer, at the surface of the airfoil. If you know the pressure distribution at the surface of the airfoil, all you need to do is integrate around the airfoil to get the lift. Note that this entire process uses only potential flow (inviscid, incompressible, lossless) and the bernoulli relation, but it is very accurate at describing the lift generated by an airfoil at high reynolds number below mach 0.3.
I am not sure if you deliberateness did this - you basically described the algorithm used by XFOIL.
To answer your final sentence, the Kutta condition applies to pretty much every flow around a non-stalled airfoil. It isn't purely a theoretical construct - if anything, it's more of an empirical observation describing the behavior of flows around objects with a sharp trailing edge. By itself, it doesn't actually serve much of a purpose, but when combined with other fluid dynamic principles, it can be quite useful in determining the properties of the flow around an object.
I answered your first sentence by noting that the Kutta condition doesn't apply to separated airfoils. The Kutta condition, to be exact, is a mathematical boundary condition based on empirical observation for nonviscus flows. Flows modeled with viscosity do not need to satisfy the Kutta condition because it does not need to be employed. The Kutta condition is not needed to solve the potential flow field, however it is needed for a nontrivial solution (IE the real world flow).
Also, I would like to show you why the Bernoulli equation does not correctly explain why lift is generated on a airfoil. Yes, it can be used as a mathematical approximation to a real flow, but it is a very misleading statement from a physical standpoint. Here is an example:
Assume that the lift generated by an airfoil can be described entirely by the differences in velocities on the upper and lower surface. Let's also assume that we are flying at sea level under normal conditions. The equation describing the lift due to the average velocity gradient will be given by
1/2 \rho (V_{top}^2 - V_{bot}^2) = L/S
ρ = air density
L = Lift generated by the body (lets say wing for simplification)
S = lifting surface area (wing area)
V = velocity
The equation will become with some substitution:
V_{top}^2 - V_{bot}^2 = C_L V^2
Assuming that the reference velocity for the lift coefficient (freestream) is equal to the velocity under the bottom half of the wing:
V_{top} = \sqrt{C_L} V_{bot}
It is clear that when the lift coefficient is low (low angle of attack, flying a high speed with large wing area, cruise conditions) that the bernoulli principle describes lift sufficiently well. However, let's consider take off or landing conditions when the lift coefficient is large. For a specific example, consider a large airliner landing at 160 mph (74 m/s) with a lift coefficient of 4. The average velocity above the wing will be:
V_{top} = 2*74 = 148 [m/s]
If the lift on this wing was due entirely to the differences in velocities above and below the wing, then the mach number at the top surface will be .43, and the bottom will be .215. It doesn't make physical sense that the mach number above the wing will be nearly twice the free-stream speed. Lift, therefore, cannot be described by the bernoulli principle alone. A better explanation is the conservation of momentum.
SIDE NOTE:I wish I could find better/specific numbers, but these are ball-parked estimates from the internet. I know that triple slotted flaps, used on some large airliners, can have maximum lift coefficients around 6. Specifics are proprietary information sadly.
