Does this exist: instrument that measures voltage at many points on a pcb?
The device you are imagining is called a bed-of-nails test fixture. However, the soldermask is not typically pierced; instead, test points (pads with no component associated) are added to the PCB design as needed.

The “needles” are pogo pins, contacts designed for the purpose consisting of an appropriately-shaped tip (which may be sharp, merely conical, concave, etc. depending on whether it is touching a pad, hole, or already-installed component) backed by a spring to provide consistent pressure for each individual contact.

A related device specifically for bare PCB continuity testing, which is necessarily slower but does not have to be made to match the specific board design, is the flying-probe tester, which has two or more XY positionable contacts which are operated by computer control to sequentially test for continuity (or lack thereof, as appropriate) between many pairs of points on a board.