As the old quip goes, I would write a shorter letter but I don’t have enough time. In this case, I’d make an animated gif but it’s late here, so I’ll have to settle for a bunch of words. My apologies if you are a visual, not a verbal learner!
I assume you haven’t used components because you are new to SketchUp and not yet comfortable with them? I’d strongly recommend that you study them because a) they will make your table model much easier to manage, and b) they will reduce or eliminate some issues you are having. The SketchUp tutorial videos have sections about components and groups.
Alas, at this stage, it might be easier to start over than to try to separate the bits that compose each such component in your model, as they are all stuck to each other.
One strong argument for using components is that your table has repeated, identical elements. The four legs are the same, just moved and rotated. The two end aprons are the same, as are the two side aprons. The top is logically a central panel with two identical end rails and two identical side rails. The side parts are essentially the same as the end ones, just lengthened, which can be accomplished by making a copy unique and the editing it. Making components and placing instances with appropriate location and rotation will let you edit just one and have all the others duplicate the edit automatically.
For example, if the end rail of the top is a component, you can cut half the groove out of its edge, and the same groove will appear in both instances. When you have all edges of the rail appropriately shaped, you can make a copy of the component, make it unique, open it for edit, and drag the bits of one end out longer to create the side rail component.
Components are also essential for the “Dave method”.
You appear to be using Layers in an effort to sidestep components. But that won’t work! In SketchUp, all edges and faces interact (stick to and intersect) with each other regardless of whether they are visible or not, and all a SketchUp Layer does is to switch multiple components between visible and not with a single setting. To prevent geometry from interacting, you have to isolate it in a separate “context”, which is what components (and groups) do.
When you want to use “intersect faces with” on a component, you need to open the component for edit and put a copy of the “cutter” into the same component as the rest of its geometry, else the intersection faces will not be in the component. You can initially draw the cutter outside the component and place it appropriately, then use copy and paste-in-place to get it inside the component for the intersect. By doing it that way you can use the same cutter for both of the components that meet at each joint, for example, the end rail and the top panel.