But as soon as the enemy gets a side, dorsal, or ventral shot in at you, you're screwed.
Not really. The spindly Vulcan shapes minimize side profile as well, if not in total area then in terms of the maximum dimensions of the area - they're harder to hit than identical-volume spheres no matter how you look at it. A spindle is a good shape no matter whether you fly it point first or sideways, but the former a) looks good and b) helps you more in fights you
The Klingon ships remain spindles in side view, even though their wing structure presents a big target to dorsal and ventral threats. Many Starfleet vessels have the same dorsal-ventral weakness, due to their saucers and sometimes their engine pylons. Why is that? One could argue that ships at warp tend to conform to local subspace fields or something, and hence maintain like orientation. But most fighting takes place at sublight, and there any orientation ought to be possible. Why have the vulnerable saucers or wings rather than smaller-target-area cylinders or spheres or spindles?
Well, the Feds might trade combat prowess for other things such as comfort - big broad decks might be much more useful than lots of smaller ones, or decks curved like onion shells inside a sphere. Klingons in turn might want to have actual wings for atmospheric lift, as they take such a holistic view to warfare...
Not that the Vulcan warp rings should make for poor aerodynamic surfaces, though. Ring wings have many advantages and have been used in guided bombs for a century now. Perhaps the Vulcan warp mechanism does double as a lifting surface on occasion?