The sticky foam is there to prevent mechanical buzzing from an imperfect junction between the guitar body and pickup chassis. I was working on an older Teisco "Tulip" guitar the other day, and the pickups were also of the form that screws into the surface. Except in this case they sit on top of the pickguard, rather than affixed to the body directly.
I suppose one could approach your guitar in two ways. One is to preserve the guitar in the spirit in which it was created by finding updated surface-mount pickups that can fit the available space (and body-to-string clearance is critical here), and turning it into the best possible version of itself.
The other approach is to view it as a platform for development, maybe cut a hole or two, and install pickups better suited to whatever you view as "jazz" tone. Twenty-five years back, I had a Pyramid semi-acoustic that I modded before selling it. Much like what I imagine your Raven to be, it was fully hollow, which made it prone to feedback at anything greater than speaking volume, unless you were quite distant from the amp. Some more professional guitars (e.g. Epiphone Casino) are also fully hollow, but tend to use materials, like thicker wood for top or back, that diminish susceptibility to feedback.
I mention this because if your intent is to make it more than something you strum unplugged, you may want to consider ways to reduce susceptibility to feedback. That could be achieved by installing some sort of post or small block inside the body; ideally under the bridge. And that, in turn, would be facilitated by cutting one or two holes in the top for installation of humbuckers or P-90s. Alternatively, some sort of dense foam block could be wedged inside; not visible from the outside, but still damping unwanted resonances.
The limiting factor there may well be the nature of the neck/body joint. The picture suggests there isn't much room for any sort of larger-footprint alternative at the neck-pickup location. But at the same time, it also suggests a very normal bolt-on arrangement, given the extended heel. If you are amenable to it, take the neck off, post a pic or two of the lay of the land at the neck/body junction, and perhaps that will prompt some useful and feasible ideas. At the very least, one could pop a hole near the bridge for a larger-footprint bridge pickup. That said, I could see where one might want to preserve the cosmetic aspect of same-pickup-style-at-bridge-and-neck.