That vehicle is anything built before the ~1970s. They're plentiful, and not many people want them. :-)
(I don't believe that there's a demand for a modern, bare-bones car. There's a demand for ever cheaper, air conditioned cars with power windows, intermittent wipers, power steering, etc. though)
There are more reasons to not want one of them than simply the lack of conveniences ;)
I mean, most of my experience is working on fuel-injected stuff... but have you ever tried to rebuild a carb? it's damn difficult to get right, and there's no computer telling you when it's correct; you have to essentially guess.
The first issue is simplicity and access. Old cars have big engine bays with everything spread out, easy to see, easy to get at. There are very few wires and it's pretty obvious (to a gearhead) what each component, wire, hose, etc... does. New cars have way more "stuff" packed into way less space. You can't see most things, and can't get at most things. There's a million wires, mostly run through harnesses, and plugged into various computers, etc...
Rebuilding a carb, while not easy, is easier than trying to diagnose if your poorly running engine is because of: bad MAF, bad O2 sensor(s) (some cars have 4-6 of these), bad O2 sensor wires (good luck tracking those through the various wire looms, etc...), bad computer(s), bad injector(s), bad FP, etc... Troubleshooting a throttle cable is easy compared to issues with a drive-by-wire system.
New cars are safer, perform better, handle better, etc... but working on them is a pain.
Coming from an (amateur) electronics background, I can say the mechanical aspects seem just as opaque and confusing.
For example: Most electric systems can be tested with a multimeter and needlenose pliers. Repairs don't get more complex than soldering.
There's a garage full of specialized tools needed for the mechanical side of things. Checking fuel pressure, removing tires, etc.
I will not argue that fixing the main relay on my father's 1962 Thunderbird was way easier (I won't even get into the difference between dropping the fuel tank vs. a newer car) I believe for the most part newer cars are not much harder to work on. I really like what Volkswagen did with their VAG-Com stuff, really makes troubleshooting easy...like stupid easy, plus it interfaces with the laptop for datalogging.
That's the best part of mechanical components for me: When it fails, often they just need to be cleaned.
1986 and 1990 Honda Civics: a PITA: just no space to turn a wrench in the engine compartment. Replacing a radiator took an absurd amount of time (and oh my aching back!)
1998 Toyota Sienna: replacing the radiator (w/engine cold) took 30 minutes the first time, 15 the second and third. My 13yo son displayed some interest so we did it an extra time :-). Changing the rear-cyl-bank O2 sensor (wedged against the firewall and crossmember) was a PITA.
1995 Camry w/same V6 engine as above Sienna: changing the same rear-bank O2 sensor was a breeze (bigger engine compartment).
Access is the key.
BTW I own two 1973 Toyota trucks, bought because they're simple to work on and exempt from CA smog testing (points ignition might survive the anticipated solar EMP!). And I agree that working with carbs is a black art with minimal feedback available, and EFI (real-time feedback!) is a major advance. Maybe EFI engines will "find their way" into these trucks :-)
During the first .com, I owned a '92 bmw 325is. Fun car, but don't take it to a mechanic. I mean, parts weren't /that/ much more expensive, but mechanic time and parts markup was, I assume for the "well, if you can afford a bmw, you can help pay for my kid's braces" effect.
First I got the Chilton's brand "book of lies" and yeah, figuring out how to do even simple mechanical things was pretty difficult. I ended up wiring a bunch of stuff in with 24awg wire.
Later I bought the 'bently publishing' manual for the thing, and god damn. everything was really simple, and it all fit perfectly.
I tell mechanics around me this all the time (North Idaho)...I personally can rebuild and troubleshoot fuel injection with no issues, I have no idea how to tune a carb. Why? Simply because I haven't had to do that yet. People seem to be scared of anything with a computer, its always the ECU they replace first for some reason...usually the problem is a sensor or vacuum leak (from my experience with things).
Older cars are easier to work on because you typically have more room and fewer custom tools, plus you need very little electronics knowledge. I was an electronics tech before I started working on my own cars so I guess that helps.