No, Schwarzschild black holes probably do not exist. We expect astrophysical black holes to be Kerr black holes, and we expect that most of them have a lot of spin. As the diagram at the end of this answer shows, supermassive black holes generally spin at relativistic speeds.
Stellar mass black holes are formed in core-collapse supernovae. They can also form when a neutron star collides with its companion (which could be another neutron star or a normal star); neutron stars are also formed in core-collapse supernova events. Collapses and collisions (of course) conserve angular momentum, and many young neutron stars are pulsars, spinning many times per second. However, some of that angular momentum may be carried away by the ejecta of the supernova explosion. It appears that core collapse can be highly asymmetric, which can give the remnant considerable proper motion, a phenomenon known as a pulsar kick:
A pulsar kick is the name of the phenomenon that often causes a neutron star to move with a different, usually substantially greater, velocity than its progenitor star.
The cause of pulsar kicks is unknown, but many astrophysicists believe that it must be due to an asymmetry in the way a supernova explodes. If true, this would give information about the supernova mechanism.
It's not easy to detect an isolated inactive black hole, or to determine its angular momentum. And if the black hole is active, the accretion disk will have high angular momentum simply due to its orbital speed, even if the spin of the black hole itself is relatively slow.
So why do we study Schwarzschild black holes? For the same reason we study Special Relativity even though spacetime is generally not flat. You need to thoroughly understand flat spacetime before you try to learn General Relativity. And you need to understand the Schwarzschild metric before you add the extra complexity that spin brings to the picture.
Besides, the Schwarzschild solution is a useful model for any spherical body with relatively low spin, it doesn't only apply to black holes. Thus you can use it (for example) to calculate the gravitational time dilation on the surface of the Earth.
As jawheele mentions in the comments, real black holes aren't exactly Kerr black holes either. The Kerr solution, like the Schwarzschild solution, is an eternal vacuum solution to the Einstein Field Equations. And we don't expect real black holes that formed through astrophysical processes to be the multi-universe gateways that the Penrose diagram indicates.
Bear in mind that we need a quantum gravity theory to properly talk about what happens at the core of a black hole, and even with such a theory we cannot empirically validate its predictions directly because we cannot extract information from the other side of the event horizon(s).
