Where does the flat plate theory in this paper come from?
Ludwig Prandtl's lifting line theory produces a Taylor series as the result of the lift equation, and in almost all books this has been simplified just to the first term of the series, which then was even robbed of its sinus function to simplify it even further by using a small-angle approximation.

What you will find in books today is something like: $$c_L = c_{L\alpha}\cdot\alpha$$

$c_l$ is the lift coefficient, $c_{L\alpha}$ is the lift curve slope and $\alpha$ is the angle of attack in radians.

What the author should at least mention (but I think most are ignorant of this detail) is that a more precise version would be: $$c_L = c_{L\alpha}\cdot\sin\alpha$$

And now the books should tell you that there are more members in the solution to the lift equation; the one above is only the first one which dominates the result for small values of $\alpha$. But they don't. That is OK when the aircraft moves only through the small angle of attack region of attached flow, but for the perching motion you need to look at bigger angles, so the small-angle solution will produce noticeable errors.

The lifting-line theory assumes inviscid flow and does not know about flow separation. Perched flight of birds would be impossible without the high angle of attacks which produce flow separation and makes use of instationary effects which delay separation. All this is not covered by lifting line theory, but nevertheless it produces quite useable results. I guess, however, that the equation in the paper was found empirically.

The lift equation cited in the paper is a consequence of observing forces in a 360° polar, one where the airfoil is rotated through full 360° instead of the narrow range between maybe -8° and +12° where flow is attached and lift forces vary linearly with the angle of attack.

The plot below is from Hoerner's book "Fluid Dynamic Lift" (Scribd) and shows the results of several measurements over 180°. Apart from the spikes around 15° and 175°, the forces correspond well with $c_L = x\cdot\sin\alpha\cdot\cos\alpha$, where $c_L$ is the lift coefficient, $\alpha$ denotes the angle of attack and $x$ is a proportionality factor (the paper postulates $x$ = 2).
$c_L$ over 180° angle of attack from Hörner&#39;s book">

A flat plate has no leading edge radius and, consequently, no leading edge suction, which would be responsible for the aforementioned spikes. Therefore, the simple trigonometric approximation will fit the data quite well.

Both the flat plate and Newtonian theories assume the aerodynamic force to be orthogonal to the plane of the wing or plate, and when you take the portion of that force which is orthogonal to the direction of movement, which is how lift is defined, you get above trigonometric equation. This is simple geometry and shows no deeper insight into fluid mechanics.