Basically designed as the submarine equivalent of an interceptor, get somewhere fast and open fire.
IIRC the top speed of the alfa was above the speed of the NATO torpedoes in use at the time.
https://en.wikipedia.org/wiki/Alfa-class_submarine#Impact
Like most fears over high-performance Soviet hardware though, the threat turned out to be overblown compared to the actual numbers build / operational state.
BTW the density of the sub is also for the most part pretty much equal to the density of water.
The only times where it's going to be either greater or lesser than water is when it's doing a static dive or fast ascent.
When the sub does it's initial dive and takes on water into it's buoyancy tanks it becomes about the average density of salt sea water once that is achieved it can use it's diving planes (foreplanes) to adjust it's depth.
The soviets also had to use titanium in many applications that would require high end steel as they both had plenty of it (almost half the world supply at the time) and their steel industry was decades behind the west and they just could not make the required quality of steel alloys. The soviets ended up using titanium as a replacement and became quite good at creating titanium alloys and working with titanium in general the US would most likely not be able to produce a hull out of titanium (at that time) without investing way too much effort into building that expertise which would be pointless given the titanium resources they had and their ability to make steels which were as good or better for most applications. The soviets had to resort into using titanium for every thing from infantry armor plating to razor blades.
Basalt fiber is similar to e-glass fiber, but has greater stiffness.
I'm guessing they made a few small fiber-wound spheres and did a crush depth test on them, but the results discouraged further experimentation.