I just installed these under some cabinets, and they work great.
Most LED under cabinet lights are similar so... Each "strip" is made to chain to the previous strip. In my installation, the first strip is plugged into an outlet with the large transformer you have pictured, and then a 12v jumper goes from the first to the second, second to the third, etc. I have 6 lights in a chain.
You installed 5x 120v drops... You could have a transformer at location #1, #2, and #5, and chain #2>#3>#4 and #5>#6. For each separate 120v connection you will need a duplex outlet and a 12v transformer. For the lights that I linked, the barrel connector on the transformer needs an adapter to fit the smaller barrel of the lights, so pay attention to that as well.
Amazon sells the transformers and jumper wires separately, so you can buy them as needed.
Also note that your linked question is dealing with RGB "strip lights" that are cut-to-fit, so it's much more complicated than plain'ol white, modular LED lights.
You are conflating 12V systems with line-voltage systems
When you say "hardwired lighting", you are actually referring to line-voltage ligthing. This runs on 120V or 230V (actually, it runs on ~12V but has an onboard conversion module) and must be cabled to the high standards of the electrical code for mains wiring. This stuff can kill you; if a lamp falls off and drops into a sink of dishes you are washing, you get to see how good your safety protection is on the circuit.
When you say "plug-in lighting", you are referring to one format of low voltage lighting. This runs on 12V or 24V, and the electrical code for wiring methods is much more relaxed. This stuff cannot kill or even shock you. If it falls into a sink of dirty dishes it will probably remain lit.
Low voltage lighting is better - for safety, for ease of wiring, and for cost - not needing a high voltage conversion module in every light.
It would pay to do some more learning about these two very different technologies.
One neat thing about low-voltage is dimming works much, much better in terms of which lights they can dim, and how much they can dim. That allows you to over-lamp your installation, and back down the brightness to what is comfortable at night, while being able to pump it up for working by day.
Convert this entire system to low voltage
The problem is, the wiring methods here are a nightmare. One does not simply put mains wiring through a crevasse in the wall. It is totally unacceptable as 120/230 mains wiring and must be completely redone at considerable expense, with lots of drywall work.
On the other hand, this sort of thing is typically acceptable for 12V/24V wiring.
Plug or hardwire has no bearing on voltage
Plugs or hardwiring are a wiring method, and work on either one. That means you are free to use hard-wired wiring methods on 12V/24V stuff.
Best Answer
Here is what I would do - install a junction / receptacle box in the wall - run your wires through it - install a strain relief wire Tie on the wires to keep them from exiting the receptacle box or use the built in relief. That will keep your wires from falling back into the wall.I would then place my piece of dry wall over that receptacle box to make it look nice again - so all I have is my wires sticking out.
You could manage to hide the wires by coming through your cabinet just above the bottom seam of the cabinet and then immediately out the bottom to your under cabinet lights. or a simple notch in the bottom cabinet lip to feed your lights.
Those wires being as they are and also in the wall like that - perhaps even go to one of those switches are PROBABLY 120VAC - PROBABLY..