Class is a reliable predictor, there's just a terrible understanding of what class
is - probably because most modern economics degrees don't cover economic history and focus just on models of individuals and not collectives. E.g., the NRS' definition of social classes (the ABC1C2DE system) includes people receiving the state pension, since they're not in work. But you can be receiving the state pension and be rather wealthy indeed, so it throws the entire distribution out of whack.
The other terrible thing you see people to do is try and do regression analysis on income and make conclusions about class based on that. That means you end up grouping the fresh-out-of-university undergraduate earning £18,000 as part of her first job but who can rise to a management position earning £40,000 within 5 years together with a 55-year old who used to work in the steel factory but got let go and has taken up a position in retail with no career prospects at £18,000. They're very obviously not in the same class.
A class is a group of people with similar relationships to the means of production, which in simpler terms is something like: they can do the same jobs and can expect to have very similar career trajectories and earnings - they exist in the same slot of the economy. One of the best ways of determining someone's class in the modern world is... their educational attainment! Having a degree means you have access to an entirely different set of jobs that can lock out those without degrees. So saying that "class has no correlation but education does" makes no sense - education has an effect entirely because it alters your class (alters your economic potential/your position as a cog in the economic machine).
That does mean there is no 'working class' - that's a lazy term, albeit a convenient shorthand. You have 'the working classes' - because a factory worker doesn't necessarily have the ability to transfer into nursing; they demand different skills despite both being at the bottom end of the income-ladder - and a pensioner is in an entirely different class to the factory worker even if they're both equally poor in terms of income, because one is a worker and the other is dependant on the state, which are not the same economic position. So you need to have much more careful examinations than we seem to get, unfortunately.
Of course, the depressing thing is that once you realise this, you also realise that the Conservatives are relatively more working class (in terms of support) than Labour is - a greater proportion of Labour's supporters are of the middle classes. And that's Labour's problem! Too many educated university students with degrees, not enough people who dropped out of school.
This is happening everywhere, incidentally - the Republicans are now the party of the American working classes. The Front National had a plurality of the French working classes. It's new and terrifying and completely at odds with the historical precedent.