A lot of the issues surrounding split tickets were recently covered
here.
I'm not sure that split tickets are either 'good' or 'bad'. Some of the reasons behind them existing are good; some are bad.
The network is complicated because it's so varied: if you can take the Esk Valley line, the West Coast Mainline, the London-Brighton line, the Cambrian Coast and the Wirral Line and design a simple ticketing system that could cover the different speeds, flows, demands and capacities that cover all these lines, then you ought to be working for NASA or GCHQ, not some train company! However on top of this complexity, we also add multiple train companies, 'PTE' areas around major cities where fares are subsidized, mass overcrowding of commuter trains, products such as Rangers and Rovers aimed at leisure travelers, and so on. So we also add in lots of complexity too, which creates some 'bad' reasons for splits.
This is why splits exist and work. In an ideal world they wouldn't - fares would be clear, and people could easily buy the cheapest fare for their journey as a through ticket. It does seem mildly unfair that splits are only available to those with the time/capacities to explore and discover the splits, and that they're only available on certain routes for reasons which are often quite circumstantial. However, the network is complex and even if we ironed out all the complexities caused by privatization or mismanagement, we'd be left with a system that either contained split ticket savings, or over/undercharged people in certain places.
With regards to 'pitfalls' - yes, there can be, depending on how you split. Most obviously, most splits involve some loss of flexibility in the services, routes or times over which you can travel. People splitting ought to understand these when making that decision.
