How can the SUVAT equations be derived from Newton’s laws of motion?

The constant-acceleration kinematic equations are most useful as a special case of Newton's Second Law. In other words, it is incredibly common to look at situations where the forces is constant, and in such cases the solution to Newton's Second Law (i.e., how $s$, $v$, etc. behave as functions of time) is given by the SUVAT equations.

To be explicit about this, if $mathbf{f}$ and $m$ are constants (as is typically the case in introductory mechanics problems), then we have for 1-D motion $$ f = m frac{dv}{dt} quad Rightarrow quad a = frac{f}{m} = text{const.} $$ and from this point on the derivations in the threads you've linked follow.

If I understand your objection correctly, it's that we could imagine some parallel universe where the laws of dynamics were different. For example, maybe we would have $f^2 = d(mv)/dt$. In this universe, for constant force and constant mass, we would still have constant acceleration, and the SUVAT equations would still hold.

And I suppose that's true, but it doesn't mean that the SUVAT equations can't be derived from Newton's Second Law under certain assumptions; it just means that there are other sets of assumptions that would lead to the same result. To put it another way, Newton's Second Law (with constant force and mass) implies the SUVAT equations; but knowing that the SUVAT equations hold in all cases for constant force and mass does not imply Newton's Second Law.

Arguably the suvat equations are more mathematical than physical. We start with axioms – the definitions of velocity in terms of displacement and time, and acceleration in terms of velocity and time – and derive the equations. Indeed I believe that Galileo himself thought of kinematics as part of geometry. And this was more than 250 years before Minkowski!

It's possible, I suppose, that a teacher might motivate the study of constant acceleration by considering a body acted on by a constant force, such as a trolley pulled by a weight connected to the trolley by a string passing over a pulley, but I'd hope that the teacher would point out later that no specific set-up is required by the equations themselves.

The SUVAT equations are simply a mathematical consequence of constant acceleration

$displaystyle frac {d^2x}{dt^2} = frac {dv}{dt} = a$

together with the initial conditions $x(0) = 0$ and $v(0) = u$. There is no mention here of force, so they have no logical connection with Newton's Laws. If an object moved with constant acceleration because it was being propelled by pink unicorns rather than following Newton's Laws, the SUVAT equations would still apply to it.