I have a bike with Shimano 105 dual-brake/control levers. But the right one stopped shifting, so I took it to my LBS to have it serviced. I asked them to take it apart and clean/grease the mechanism. But they refused. They said "dual brake levers are not meant to be serviced" and offered to spray WD-40 into it to loosen it up. Is it true what they said? Or is it that they just don't want to deal with it? I found a video on youtube that shows how to dissemble, clean, and reassemble one of these levers, so I know it can be done. Also, Shimano makes a tool designed to help take it apart. So what's going on with my LBS?
I've successfully destroyed several right-hand levers by attempting to fix them once they've started acting up.
One was a broken basket in the shifter, and I forget the actual failure in the others. If I'd had duplicate identical shifters with different faults, perhaps making one out of two would work but cobbling was unsuccessful.
If your LBS accepted the job, they could be sinking a lot of mechanic time into the work. At least one hour and potentially up to four. At $100 an hour you've already spent more than the cost of a new part, and there's no guarantee they can fix whatever's wrong inside.
Even if they do get it working, it may break again for the same or a different reason.
Frankly its not cost-efficient. On the other hand, your time is "free" so you could totally spend an enjoyable evening working on this, and if it doesn't work then you just buy the replacement anyway.
The only time it makes sense to fix an old part is if that is no longer available. An irreplaceable vintage part, or a unique part, or something with historical or sentimental value; all those reasons can override fiscal responsibility.
As gerrit notes - ecological responsibility also might drive one to try a repair; less waste is a good thing.
And the simply joy of learning how your things work can be reward all by itself, even if the item may never work again.
The answer depends a little on the exact model and how deep you need to go.
In general the problem is that replacement parts aren't available new if something is broken, and if nothing is broken then most of the other things that could go wrong can be resolved by cleaning and lubricating them whole.
That leaves the possibility of making one good one out of 2+ bad ones. There's nothing wrong with that in principle, but when you actually try doing the work of fully disassembling them (as opposed to simple things like extracting the lever pivot etc) you see that it's a field in itself. There is no documentation and they are likely assembled with special tools and/or robots. There's also the question of what kind of warranty you can offer on the expensive, time-consuming work after one part (usually one of many tiny springs) has already broken, possibly in fatigue. You can't practically replace all of them, and not warranting the work is a recipe for bad customer experiences even if agreed upon up front. Shops are well within their right to only replace STIs, not repair. It might be helpful to bear in mind that people who work at shops usually like fixing bike stuff and are happy to do it if possible within the bounds of good business.
A small addition to this that I don't think has been fully covered:
Internal parts for Shimano shifters aren't available. As previously noted, they are also disturbingly difficult to disassemble. This also appears to be company policy with other systems such as hydraulic brake internals and seals.
If you want levers that can be serviced, try going Campagnolo. When i don't want to service these myself I can send them back to the Campag Service Centre, which is generally very cost effective for most repairs. The oldest levers i've sent in for repair were 9sp Daytona shifters from the late 1990s.
If the tool is the one I'm thinking of, it mainly allows you to take the whole main assembly out of the lever. That assembly is available, at least for some models, as a spare, costing about the same as the assembled lever (a matter of supply and demand). I've done that swap when my lever stopped shifting up reliably after shifting down.
So it's also good if you've got a working lever with a cosmetically damaged body, and a glitchy lever. You still don't have great access to the real mechanism.
To get far enough in to clean and regrease is a lot more hassle. I have opened up a (Sora rear) lever that far, when I'd screwed up a roadside cable replacement and got it jammed inside. It took all afternoon. With practice it would still take a good hour or two. And there was a very slightly off feeling to the lever afterwards. About 6 months later I replaced it as it was misbehaving. They seem to last around 3 years/20 000km for me.
The flat bar equivalent is often a little more serviceable, but still really can only be flushed with something to dilute the grease and wash some out, then oiled. Maybe a watchmaker could dismantle the mechanism enough to fully degrease the parts and reassemble, but doing so just isn't a thing. I suspect your YouTube video actually does a fairly superficial degrease, or puts the main mechanism in an ultrasonic bath, but then you have the problem of getting new grease in everywhere.