Depends on how you define 'fraught with problems.' They're old cars, and old cars have problems; and some new ones do too (I'm wondering if we'll ever get airbags that don't grenade randomly). They are, for the most part, state-of-the art technology for, oh, 1940 but are butt-simple to work on. One thing that seems to baffle many stateside mechanics are S.U. carburettors--you'll learn new words with odd spellings--which are of the 'variable choke' type, operated by one Mr. Bernoulli (a variation of which was employed on early Rolls Royce Merlin engines, of Battle of Britain fame). I believe some motorbikes use a variation of them; the venturi cross-section is changed depending on load, as opposed to American cars' with fixed 'choke,' which require several jets and circuits and may still have a 'flat spot' or two. My Honda quad has a variable choke carb whose 'choke' piston is raised and lowered mechanically. IMO, the S.U.s are a more elegant design than the fixed choke carbs--for example, the Holly 'teapot' on my dad's '55 T-Bird, which was an abomination--and, once sorted, require no further futzing for many miles. Fuel pumps have been my biggest bugaboo over the years, but can be OK if you find a good one and I can change one on the side of the road in 20 minutes (my inherited 2000 Lincoln LS seems to go through them pretty regualarly too, but it has a Jaguar engine).
You'll hear the term 'Lucas Electricals,' usually spoken in a derisive tone. The electricals can be glitchy, and are the source of much comedic material--right up there with lawyer jokes--but can be managed. I was a bit surprised that my '67 3000's wiring used bullet-type connectors identical to those used on our 2 Model As (the youngest was a 1931). These, I believe, are the cause of many of the electrical issues as they encourage corrosion and are mechanically weak. This can be mitigated by packing the connectors with dielectric grease before inserting the wiring tips and routing and securing the harness properly. Repop electrical parts are available but sometimes of suspect quality. Also, it seems to me the insulation used on harness wires is particularly thick for only 12V, and I think it gets brittle and prone to cracking over the years. The good news is the harnesses can be replaced in situ in one long day's work.
The 600lb cast iron engine can have porosity issues, but I don't think they are terribly common. Also, since it's a flat tappet valve train you need to be judicious about break-in procedures and oil types (Castrol and Valvoline 20W-50 are a couple faves).
In a word (or two), the cars are a commitment; better IMO if you can do at least some of the work yourself but many owners find a mechanic/shop they trust and that's fine too. The cars' values have dropped somewhat over the last decade or so, but owners' enthusiasm has not.