Note that the volume within the Schwarzschild radius does not increase exponentially with mass, rather as the cube of the mass. You represented it that way in the equation, but it would be good to get it right in the wording too, lest you confuse someone you're speaking with later.
Essentially, what you've stumbled across is the notion that if you had a region of matter with some average density, then there exists a radius of this region above which it would automatically be a black hole. This is true. If you have a cloud of gas with uniform density, then since the mass of this cloud would increase proportional to the radius cubed, its Schwarzschild radius would increase proportional to its actual radius cubed. While the Schwarzschild radius would start off smaller than the actual radius (thus, no problem), as the size of the cloud gets arbitrarily large, the Schwarzschild radius overtakes the actual radius and the cloud would automatically represent a black hole.
This does not mean that we need to worry about runaway black hole formation. This only holds for regions with a defined center of mass. An infinite or near-infinite region of uniform density would not form a black hole. You might disagree and call up the shell theorem. However, this does not predict a black hole because without a definitive center of mass, every point is effectively the center and, thus, the Shell theorem would state that every point feels the gravitation effect from a sphere of zero radius (and that all matter outside the sphere contributes no net force). Arbitrarily choosing a center and claiming a black hole forms around it is a mistake. It will always be cancelled by the gravitational effect of the matter found on the opposite side of an observer. I admit, space would be weird, but a beam of light would not be deflected in any particular direction by gravity; it would traverse the space perfectly fine (apart from scattering off the matter).
Since any given point in space is effectively the center of the observable universe at that point, the average mass density of space must cancel out on all sides. This means the only way a black hole could reach a "runaway point" is by accumulating overdensities of matter fast enough to get above that radius first; simply having a universe big enough to place an observer at that radius is not enough. There are two important things preventing such a black hole: first is the Eddington limit, as mentioned in one of the comments, which makes it very difficult for the black hole to accrete the mass it needs quick enough. Second is that, given the average density of the observable universe (including dark matter and not just ordinary matter, because why wouldn't you?), the Schwarzschild radius of the observable universe ends up being less than twice what its actual radius is. This is great because, as discussed you can't have a black hole form spontaneously just by having the Schwarzschild radius of something like the universe be greater than the actual radius (every point in space is the center) and you'll never have a pre-existing black hole grow to be larger than about half the radius of the observable universe because it would become out of causal contact with itself (one side of the event horizon would be at the edge of the observable universe seen from the other side). Not to mention the whole dark energy thing that has a negative energy density (but let's not complicate this further).
Short story? You're not technically wrong, but there are lots of things in place preventing what you might worry about from ever becoming a problem