The "First Equation of Motion" you define is perhaps more accurately called the "First Equation of Motion with Constant Acceleration."
One would need to use Calculus to calculate the change in velocity when Acceleration is not constant, but what you call a "varying variable."
Your first equation which you arrive at by Algebra: $$V_f = V_o +a \Delta t$$
yields the correct change in velocity because the change in velocity $\Delta V$ over any interval of time $\Delta t$, large or small, is always $$\Delta V = a\Delta t$$ because a is always the same value.
When a is a "varying variable," a is different for different times, and therefore $\Delta V$ is different for different intervals of time $\Delta t$.
We can use calculus to find what $\Delta V$ is in this case by splitting $\Delta t$ into a very large amount of small times $dt$, and adding together the many very small resulting changes in velocity $dv = a(t)dt$. We add up the very small changes in velocity $a(t)dt$ between two points in time by writing: $$\Delta V =\int_{t_i}^{t_f} {a(t)dt}$$
Where the integral symbol is an elongated S symbolizing "sum."
I hope this helps you see when one must use Calculus in Physics.