First I think it is useful to give a brief definition of displacement and distance.
Distance depends on the path which an object moves whereas displacement is the shortest distance between two points, usually initial and final positions. Displacement and Linear distance imply the same.
You have mentioned in your question that you have used equations of motion for solving questions regarding circular motion. But you CAN'T use equations of motions unless acceleration is not constant, therefore here you can clearly see $a_t$ is not constant as it changes its direction every second. You may wonder then how to solve such questions. Yes, you have to use this equation but with a slightly different meaning. What is this contradiction. Let me explain.
I wonder whether you have a clear idea about from where these equations of motion come. This is basically from the definition of acceleration. Acceleration is defined as rate of change of velocity. This can be shown as,$$\text{rate of change of velocity}=\frac{\text{final velocity} -\text{initial velocity}}{t}$$
$$a=\frac{v-u}{t}$$ where $a$ is constant. From this you can derive $$v=u+at\tag{1}$$
And from the defintion of average velocity,
$$\text{average velocity}=\frac{\text{total displacement}}{\text{total time}}=\frac{\text{initial velocity} +\text{final velocity}}{2}$$
$$\frac St=\frac{u+v}{2}$$
$$S=\frac{(u+v)}{2} t\tag{2}$$ From these two equations you can derive other equations.
Then how do you get correct answer in your question regarding circular motion using that equation? It's simply because your $a$ and $S$ do not have usual meanings here. At this situation your $a$ represents rate of change of speed and $S$ represents distance. This comes from the definition like before:
$$\text{rate of change of speed}=\frac{\text{final speed} -\text{initial speed}}{2}$$
So you may see $l$ in your equation is not for displacement, it is obviously for length. Also you can use this equation because the magnitude of $a_t$ is constant, eventhough its direction is not (because the equation we defined is not a vector equation). Thus this conclusion of yours is not correct:
we even used equations of motion in circular motion like $v^2=u^2+2a_t(l)$ where at is tangential acceleration, $l$ is lenght of arc. But displacement here is not a straight line
Hence, sorry to say, the title of the question is incorrect too.