While replacing the battery, the guy cleaned the laptop from dust. Then, he has removed and connected the WiFi cable to clean below it.
I don't see why it would be necessary to remove the WiFi antennae cables to "clean below it." The nozzle (looks like a thin red drinking straw; below left) on those air dusters are thin enough to get into the smallest crevices of your computer. However, if your tech is a "pro," he should be using a proper air compressor with the right tool - technically called a a blow gun (below right).
Could the issue be that the cable is not connected properly?
Odds are, you've correctly identified the problem.
Resetting the SMC would have no effect as it's related to power and cooling of your Mac. Things like your location, the date and DHCP have no bearing on signal strength - it's actually the opposite. If your signal strength is too low, any/all of those services could fail.
What to do next....
Return the laptop to the person who serviced it. Make sure you document the issue in email to him straight away as well.
If it's a simple mistake of not reconnecting the antennae (there can be 3 - one for Bluetooth and 2 for WiFi) then it will only take a moment to reconnect them. However, if the antenna cable(s) and/or connector(s) are damaged, the antenna(e) will have to be replaced which means disassembling the display as the cables are routed behind the LCD panel (it's a difficult and risky job).
I don't recommend you opening your MacBook and attempting the repair. If the tech damaged something, you want to ensure that he "owns" the problem and not give him an "out" where he can say that because you opened it, you might have caused the damage.
From your Update
That butt splice connections on the antennae is incorrect. Those are coaxial wires, not single conductor strands. There are two conductors - a single inner conductor and a conducting shield separated by a dielectric insulating material which is all "covered" by a protective outer jacket. Your WiFi will never work correctly with that "fix".
If you look at the connectors themselves, you'll see they're not unlike the coax cables that provide you cable TV or the antenna on your WiFi router. There's an inner socket surrounded by plastic and the outer conductive "shell."
The bottom line is that repair is unacceptable and not going to work. If replacing the display is what he has to do then that’s what has to be done. It’s an expensive lesson for him to learn, but so goes it when he decided to charge money for service he didn’t have the necessary skills to provide (my consumer protection side coming out).
Personally, I would let him know of the issue, follow that up with an email for documentation, then take to a qualified person to replace the display and present him with the bill. Tell him nicely, at first, that he should replace it. However, it could get messy. I can't advise you what to do legally, but if I personally encountered resistance to make right what someone else botched, there would be legal action being taken. I've personally taken people to small claims court over issues like this and prevailed. Documentation is key.
