Unfortunately, each piece of this layer cake is still slightly imperfect and not totally flush after gluing.
If you're using templates correctly this shouldn't really occur. Each one should be a perfect or near-perfect copy of the others. This suggests a few possible issues with the way you're routing, as does the evident scorching* and the scalloping visible in the photo when viewed at full size.
Apart from purchasing yet another expensive specialty router bit, how can I make this "thick" stock totally flush once more?
You can do this entirely by sanding, and entirely by hand sanding if you had to. Working manually even doing it the right way (starting with nothing finer than 80 grit) it will still take quite a bit of time, but it is doable. So if you choose to go this route settle in for a long job, put your head down and work through it. It won't seem like it while you're sanding but you will eventually get it done!
Rasps and/or files
A good rasp or a suitable file will do the gross smoothing faster, with sanding or scraping to finish off. Because you'll be forced to do some rasping/filing across the grain watch out for spelching from the far edge.
Planing would be the best way to smooth off most of the surfaces here because it's fastest and most efficient, plus no dust, but unfortunately it's not possible to do the whole job this way because some curves are too tight. Additionally the concave portions require a compass plane which is a speciality plane and often not cheap (vintage ones in even OK condition command high prices). The flats and convex parts though can be dealt with using any standard flat-bottomed plane. In a Comment you mention you have a block plane and that can be used for this. Make sure the iron is well honed and close the mouth up to help reduce tearout.
Regardless of the plane used expect to have to sand or scrape to finish off.
This is what I would recommend.
Use a plane for every surface it's suitable for, and rasps/files or sandpaper wrapped around dowels or shaped sanding blocks for all the rest. Then sand or scrape to blend the areas and unify the surface.
*About the scorching, some woods are particularly prone to this and even with a sharp bit and good technique it is normal to expect to get some. But too much of it is seen particularly when you push the bit too hard or (as odd as it might seem) you're using too low a speed and/or moving the router too slowly. If you play around with feed rate and speed combinations you should be able to reduce it to a minimum even in a scorch-prone wood. And this sweet spot will also produce a better, more even routed surface (i.e. minimal or no scalloping).
Additionally, one of the secrets to getting a really nice surface when routing profiles is to do on last, very shallow, pass. One way of achieving this is to stick a layer of tape along the edge of your template (or the router's fence) for the main cut, then take it off for a final pass. With the tape removed the bit will skim off a thickness of wood equal to the thickness of the tape. This technique virtually guarantees a tearout-free, burn-free surface and if your bit is good and sharp it may be so good that no final sanding is required to smooth off. You may wish to sand to equalise how the surface takes stain or finish however.