The dark matter has to have a roughly spherical distribution and a smooth radial distribution if it is to account for the dynamics of gas and stars in our and other Galaxies.
The exact shape of any required dark matter halo is still the subject of debate and research. Whilst computer simulations of galaxy formation predict mildly non-spherical or triaxial profiles (e.g. Kunting et al. 2018) and some stellar kinematic data appear to confirm sphericity (e.g. Smith et al. 2009 ); others disagree and suggest there is some significant prolateness ( Bowden et al. 2016) or oblateness suggested by large scale surveys (e.g. Loebman et al. 2014).
Any concentration of dark matter into a disk is totally ruled out by observations of the dynamics of stars above and below the plane.
e.g. The latest word, using Gaia DR2 data, is that there can only be about 10% of the local mass density in the form of dark matter of any kind (Buch et al. 2019, agreeing with lots of previous work. For instance this was established fact at least 30 years ago (Kuijken & Gilmore 1989).
The amount of dark matter required at different radii (assuming Newtonian gravity is the correct formulation) is deduced from the rotation curve. The image below, courtesy of Nick Strobel at www.astronomynotes.com, shows a sketch of the current thinking on how much dark matter needs to be at what radius (labelled as "Corona") and takes the dark matter to be distributed spherically symmetrically.
Dark matter already "outweighs" luminous matter by 2:1 at the solar radius, yet there is room for only 10% of the local density to be in a "dark" form of any kind.
Brown dwarfs are not actually "dark objects" that cannot be found. They emit infrared light and surveys near the Sun reveal that stars outnumber them by 4:1 and that they contribute less than a few percent the local mass density.
They can also be found via gravitational microlensing, in surveys that have been performed towards the Magellanic clouds (sampling the halo) and towards the Galactic bulge (sampling the inner disk). In neither case has evidence been found for huge numbers of brown dwarfs (or cold white dwarfs or black holes) and these have been ruled out as significant contributors to the Galactic dark matter problem (e.g. Pietrzynski 2018). In particular, the numbers of microlensing events towards the Galactic bulge is perfectly consistent with the brown dwarf/star fraction in the local neighborhood (Niikura et al. (2019).
As a final remark, dark matter (whether in the form of brown dwarfs or non-baryonic particles) doesn't clump very easily because it cannot readily dissipate it's kinetic energy. There is little evidence for any clumping of dark matter on scales smaller than dwarf galaxies. In particular no evidence for dark matter is found in globular or open clusters of stars.
The stellar disk forms because the stars formed more-or-less in a disk that had already formed from dissipative baryonic gas that had already collapsed into a disk. As brown dwarfs must also form from baryonic gas then there are really only two alternatives for their distribution. Either they follow the disk and should be associated with disk stars or they should be very old and associated with halo stars that formed in a spherical gas cloud before it collapsed to form a disk and therefore still have a spherical distribution.
The dark matter needs to be much more extended than the luminous matter of the Galaxy. In neither scenario described above do we end up with tens of trillions of brown dwarfs (for that is what would be required) distributed a long way outside the solar circle and unassociated with any luminous matter.
