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One ought to be able to prove this for most braid groups in a similar way to $B_3$. It was shown by Venkataramana that Burau representations of braid groups are arithmetic in the appropriate range. Arithmetic groups should have lots of congruence quotients which are perfect by the strong approximation theorem. But I don’t have quite enough knowledge of the appropriate group theory to complete this line of argument.

This should follow from the approach to prove Theorem 1.2 of Masbaum-Reid. (Alan Reid suggested this to me)

A’Campo showed that the finite symplectic group $Sp(2m,p)$, $p>2$ prime, is a quotient of the braid group $B_n$ for some $m$ depending on $n$. Hence the finite groups $PSp(2m,p)$ are quotients of $B_n$. These groups are simple non-abelian hence perfect for most $m, p$.

A’Campo, Norbert, Tresses, monodromie et le groupe symplectique, Comment. Math. Helv. 54, 318-327 (1979). ZBL0441.32004 MR0535062

One ought to be able to prove this for most braid groups in a similar way to $B_3$. It was shown by Venkataramana that Burau representations of braid groups are arithmetic in the appropriate range. Arithmetic groups should have lots of congruence quotients which are perfect by the strong approximation theorem. But I don’t have quite enough knowledge of the appropriate group theory to complete this line of argument. https://doi.org/10.4007/annals.2014.179.3.4

One ought to be able to prove this for most braid groups in a similar way to $B_3$. It was shown by Venkataramana that Burau representations of braid groups are arithmetic in the appropriate range. Arithmetic groups should have lots of congruence quotients which are perfect by the strong approximation theorem. But I don’t have quite enough knowledge of the appropriate group theory to complete this line of argument.

This should follow from the approach to prove Theorem 1.2 of Masbaum-Reid. (Alan Reid suggested this to me)

$B_n/Z(B_n)$ is perfect and is the subgroup of the mapping class group of the $n+1$-punctured sphere fixing the point at $\infty$. Hence it is residually finite (and infinite), so it has infinitely many finite perfect quotients.

One ought to be able to prove this for most braid groups in a similar way to $B_3$. It was shown by Venkataramana that Burau representations of braid groups are arithmetic in the appropriate range. Arithmetic groups should have lots of congruence quotients which are perfect by the strong approximation theorem. But I don’t have quite enough knowledge of the appropriate group theory to complete this line of argument. https://doi.org/10.4007/annals.2014.179.3.4