It depends a bit on the scale on which you are looking. If you’re thinking of, say, the satellite before it disintegrates, or its center of mass, then, yes, this hits the ground at a shallow angle, because the satellite is traveling downward fairly slowly, no more than a few hundred miles per hour, while its horizontal velocity is very substantial, starting off at 15,000+ miles per hour, and slowing considerably as it enters the atmosphere.
On the other hand, the satellite is expected to break up, and each individual piece will pursue its own trajectory. The pieces will likely initially have very similar horizontal and vertical velocities as the center of mass. However, it is likely that, particularly for the smaller pieces, the horizontal velocity will decrease much faster than the vertical velocity (because air resistance grows as a strong power of velocity, e.g. the air resistance at 5,000 MPH is much more than 5 times as much as the air resistance at 1,000 MPH, probably 25-200 times as much).
If the horizontal velocity is shrinking faster than the vertical velocity, then the angle of the fall relative to the ground will steepen. If there is enough time, meaning the piece is small enough, then the horizontal velocity will disappear entirely, eliminated by air resistance. The vertical velocity remains of course, since it is being sustained by gravity, and in this case the piece will finally hit the ground vertically.
In short, the angle at which pieces hit the ground depends generally on their size, and the smaller the pieces, the more vertical the impact.
In case you’re wondering why air resistance acts more strongly on the smaller pieces, the reason is that the magnitude of the air resistance force goes as the surface area of the piece, which goes like the diameter d square, d^2. However, the mass of the piece goes like the volume, i.e. the diameter cubed, d^3.
Newton’s law saws the acceleration a = F/m, or in this case like d^2/d^3 = d^-1. That is, the deceleration of the piece due to air resistance is inversely proportional to the size of the object: smaller objects decelerate faster.