Is there a way to create an effect like this without applying the first 2 modifiers?
I tried to just stack all of the modifiers, but I didn’t get the same end result.
If modifiers are set exactly the same, same modifiers, same order, same values : end result should be the same.
Mentioned modifiers should not behave differently, by taking into account same geometry with applied modifiers or not applied ones.
There are 3 types of Smooth modifiers. Maybe, you did not use the same.
Maybe you forget an adjustment, a scaling of object. Or some setting of applied modifiers, used texture were different than the ones used by copy without applied modifier.
Maybe, you are not looking at geometry from same viewpoint.
There would be a difference, if you did not apply modifiers in order of their effect.
There would be a difference, if a modifier had its effect disabled for render and viewport.
Your render seems perfectly feasible with mentioned modifiers , without applying them.