You are never going to eradicate split ticketing for as long as paper tickets exist. It doesn't matter what system you have, there will always be anomalies like you mention. Split ticketing would be a great deal less attractive if you had to activate your ticket on a reader at a station. Or if your ticket system was PAYG like Oyster. There may even be ways of storing or paying for tickets that we haven't even thought of yet.
I'm not going to speculate what the fares might be for certain flows, but the answer to some flows might indeed be to double the fare, or more! I am entirely comfortable with passengers combining singles between peak and off peak to save money, because it is true that some other fares may rise quite a bit.
Hard choices. In the 1960s whole railway lines were entirely closed, yet the railway as a public service continued to function and became more efficient. It may be that certain lines like the Esk Valley line become uneconomical to run, and significant change or even closure may happen.
It would be a hard sell, you're right.
To eradicate split-ticketing through the eroding of passenger rights is effectively to say "we know you can save money by doing X, so we're going to stop you doing X". Not particularly compelling from a voter's perspective.
Smart ticketing would, in any case, be compatible with split-ticketing (such as combining peak and off-peak fares) if, as is already the case with m-tickets, all passengers need to do is activate the appropriate ticket before departing from the origin station on that ticket.
If I was a civil servant in the rail division of the DfT and aimed to achieve the government's financial objectives, increasing fares to the level at which the maximum revenue is generated, regardless of whether patronage in terms of passenger numbers decreases, would be my goal. But I would appreciate that this is not politically viable and that reducing rail subsidy, while one of the government's considerations, is not the government's sole consideration.
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I'd replace them with a season, a peak single, an off peak single (valid after 0900/0930 whatever), and the TOCs may set whatever advances they like. The only permitted route would be the most direct one, up the WCML proper. So, if LM want people to go via Northampton they'll have to set an Advance fare to do so. Yup, means less choice and some loss of flexibility. Hard sell.
I'd like that, other than the removal of the routes via Northampton (and Coventry). What if Rugby to Milton Keynes avoiding Northampton was determined to be the more expensive route? And even if the route via Northampton was, remarkably, determined to be the more expensive route, would passengers holding walk-up tickets be forbidden from travelling on e.g. the 18:49 ex. Euston? And by removing the route via Coventry, Nuneaton to Milton Keynes commuters would only be able to arrive in Milton Keynes at 07:13 or 08:55!
Regardless of the permitted routes issue, the TOCs will be unlikely to accept simplification in the manner described (another example being Milton Keynes to London) unless the average fare were to increase, perhaps substantially.
And with respect to Nuneaton to Milton Keynes, who would set the fares? Currently it is Virgin Trains, but London Midland operate the vast majority of services between these stations. If Virgin Trains were to price the fares, passengers would end up paying intercity-level peak time fares to travel on mediocre trains (particularly first class on the 350s): there is no sensibly-timed Nuneaton to Milton Keynes VTWC service on a weekday morning. But if London Midland were to price the fares, would Virgin Trains be happy with first class fares between Milton Keynes and Nuneaton (for which there are two premium VTWC evening peak trains with a comprehensive meal service) substantially decreasing?