Yes, it's done via ROV's with manipulator arms that attach salvage cables to them. Way less failure prone than trying to bring them up with the ROV's directly because the ROV's are normally tethered via a fairly light duty tether that's connected to a base station that raises/lowers the craft. That keeps things much more maneuverable because the ROV doesn't have to drag around thousands of feet of cable, and, it provides some damping from any movement of the host ship operating it.

Additionally, the ROV can stay submerged, looking for other things to grab, while salvage cables are raised/lowered. Doubly important because those sorts of winches run at ~1000ft/hr, and the wreck is ~12.5k feet down, so over a full day for one recovery.

I have never hauled submersible wrecks from the seabed, but have recovered lots of other stuff from the same place.

Depending on the weight and bulk of the component, you'd likely do one out of three (at this depth; in shallower waters, you have more options)

a) Small-ish items are simply dropped in a basket attached to the ROV doing the recovery, then brought back to the surface with the ROV.

b) Larger items may be handled by the ROV to a basket suspended from a winch, then hoisted to the surface.

c) If the items are too large for the ROV to handle, it will be used to attach a sling to the item, which is then in turn attached to the hook of a crane/winch, before being hoisted to the surface.