The problem here is that while the escape speed is indeed $c$ at the horizon, the classical (Newtonian) way of conceiving of the significance of that speed is much different in general relativity versus in Newtonian gravity - or, actually, it can be thought of in the same way in both, but it really becomes important when you are considering general relativity.
You should not think of the escape speed as simply being a speed which only applies to ballistic escapes, but rather as a speed that signifies how hard it is to escape the gravity well of whatever the thing in question is, by any method. Namely, it is "as hard as accelerating your whatsit up to that speed", whether you actually achieve that acceleration or not: remember that when considering a "slow" climb out of a gravity well even in Newtonian mechanics, your rockets have to be firing continuously and they will use at least (and likely much more) as much energy and propellant as reaching that escape speed requires despite the slow climb.
Hence, when in general relativity you see escape velocity $c$, what it really means is "it is as hard to get away from here as it is to travel at exactly the speed of light". In other words, right at the horizon, getting away is equivalent to sending your massive spaceship at the speed of light: something you already should know from special relativity is infinitely hard. Going below the horizon, it becomes "beyond infinite" - so hard that it is described with an imaginary number, which in relativityese actually means "as hard as going faster than light" and thus "as hard as making a time machine", and therefore, you really really can't get away.
Likewise the same applies to tethers: you will have to pull on it with infinite muscle to get it just up from being exactly at the horizon, and no rope can be infinitely strong, so when lowering something, all ropes must break before the suspended objects reach the horizon.
(In a sense, you could say "gravity becomes infinitely strong" at the horizon, not the singularity, but it's better to say "gravity becomes irresistible", or that the hovering force becomes infinite, because the "strength of gravity" has other definitions that are more appropriate to the general-relativistic setting. In particular, the gravitational field must be described by a tensor, not a vector, in general relativity, and this tensor does not become infinite at the horizon, but the function mapping from this tensor field to the needed hovering force does become infinite there.)