This is not correct, and the experiment has been done a number of times with no real effect, most memorably in 01961, when Project West Ford put 480 million copper needles in 3500-km-high medium Earth orbit (MEO); but it's a little bit challenging to understand why.
The first reason is that outer space is just inconceivably big, even just the space in low Earth orbit (LEO). LEO has very nearly the same surface area as the Earth, which is to say, about 40% more surface area than the ocean. But the ocean averages 3.7 km thick, and LEO is 2000 km thick. So, roughly speaking, LEO is 50 times the volume of the ocean.
That's a lot of space to miss. There's about 15000 active satellites up there, mostly Starlink. Imagine there are 15000 ÷ 50 = 300 divers swimming at random places in the world's oceans, you have a gun that shoots magic bullets that go 7700m/s and never stop until they hit a diver, and you fire one into the ocean at random. How long is it going to take that bullet to hit someone? It traverses, say, 7700 cubic meters per second, but there are 1.32 × 10¹⁸ m³ (1.32 quintillion m³) of ocean it could traverse, and on average it has to traverse 4.4 quadrillion m³ before it hits someone.
It turns out that your poor bullet will bounce around the ocean for about 18000 years, which is the same thing that will happen to one of your ball bearings if it manages to stay in orbit.
The second reason is, perhaps surprisingly, air resistance. The atmosphere thins out exponentially but never quite stops. Most satellites are at a height where the orbits of even macroscopic multi-kilogram objects decay within a few years from air resistance unless you boost them; smaller objects like your ball bearings or the West Ford needles decay more rapidly. Indeed, all of the individual West Ford needles have already deorbited, despite being in MEO at four times the height of typical LEO satellites, though 44 needle clumps remain.
The third reason is the narrow nature of orbital dynamics. The reason you have to go 7600m/s is that you're in free fall, and you have to miss the Earth when you come down, by having moved far enough horizontally that the Earth isn't there any more. If your hand grenade sends the ball bearings flying backward along the orbit and/or downwards fast enough, they'll re-enter the atmosphere immediately. What's less obvious is that the ball bearings that it sends upwards will re-enter the atmosphere after half an orbit. Depending on how forceful your explosion is, only a narrow circle that stay in almost exactly the same orbit may survive.
None of the ball bearings will be lofted to a higher stable orbit. If they couldn't hit the Earth (perhaps because we replaced it with a very, very small black hole) and none of them reached escape velocity, they would all orbit back through the point where the hand grenade went off, once they'd finished an orbit. So (restoring the Earth to our scenario) even after 18000 years, your hand grenade will have only hit satellites at the same height where it went off in the first place, or slightly higher or lower.
The Kessler syndrome is a real, serious problem for space access, but it hinges crucially on the number of satellites up there to participate in the fragmentation cascade.