At core, SHM is the shadow of a particle revolving with omega on a circle of radius equal to amplitude. Now we say that body performs SHM either if the equation of its position makes a sinusoidal function or if its acceleration is proportional to displacement and in opposite direction.
I get how in the first way we can always equate it to the shadow of a particle revolving but in second way, i.e., by seeing its acceleration proportional to displacement how can we say for sure it has an equivalent particle shadow? Yes, if it starts from the mean position, we can make a similar shadow which starts from mean and the particle revolves with omega (according to acceleration of body and amplitude). What if the body starts with velocity v from any random distance from mean? How then can we be sure we will find its equivalence to a particle shadow?
Usually in that case, going by the questions we solved in class, we find the amplitude by equations of SHM itself. But we can't use the equation without proving SHM.
I am sure there is some obvious point here but I can't see it.