The search term is:
"PCB assembly service"
"Selective wave soldering machine"
If you need to hand solder more than 2-3 board in house, you are doing something wrong. The reason these jigs cannot be bought anymore, is because they are not relevant anymore. There are good assembly houses all over europe, solving any assembly requirements, and strange requests. I dont think it it would be otherwise in the US.
We have a wave solder machine in-house. We have actually discovered that for boards with less than a certain number of pins it is actually faster to solder them by hand (put connector on board, solder repeat) than do all of the handling related to the wave solder machine (pre-connectorize, put on tray, transport trays to wave machine, load board into wave machine, remove board, put on tray, transport back to inspection area, inspect and cleanup). About all we do is the higher pin count boards (>50 pins or so), and even they are questionable when you add the setup time for the wave solder machine itself.
Part of what is driving this is the desire to over the next year or so move to some sort of automation and eventually rid ourselves of the wave solder machine. A 10-20K USD soldering robot is not out of the question here. A $50K + nitrogen system or consumable nitrogen + lots of energy selective solder machine is much less attractive. Unfortunately, every soldering robot I've encountered needs to have the board inverted to solder the pins, and also leaves the fixturing up to the user (or have a service to build one for each board for lots of money).
So the thought process is this: If we can solve the board flip problem in a way that is inexpensive to do on a per-board basis (automatic fixturing, etc), then this both cuts down the time needed for hand assembly now, and also provides us a solution for the soldering robot when the financial numbers work for the robot. If we can't resolve the board flip problem in a reasonable fashion, then we need to buy the selective solder machine (no flip required) or just continue the way we are.
One final note: I've toured several assembly lines recently. Every one I've been into employ a connection of workers who solder connectors on boards by hand. There was one in particular I remember which had a nice shiny selective solder machine which was not turned on, and in another area, a group of employees soldering connectors. I asked about why not use the selective solder machine, and their response was that they had found that the cost of setting up the selective solder machine wasn't worth it for all but the larger runs, so they still hand-soldered connectors on most of the smaller batches. I don't remember which threshold they said, but it was well over 100 boards.