To be an HGV driver requires sufficient literacy to pass a driving theory test, pass the advanced HGV driving theory test, read and follow delivery instructions for multiple locations, read, understand and be able to run safety checks on the vehicle.
It also requires the ability to turn up to work on time, communicate with your colleagues, speak to customers/clients in an appropriate way, problem solve, make decisions, follow rules.
The boy in the programme cited was illiterate. He would not be able to become an HGV driver without learning to read a lot better. But he also would likely struggle with all of those soft skills as well.
There is rightly a focus on poor white boys' literacy levels but that alone won't fix things. If their home environment/culture doesn't encourage/develop their soft skills - or possibly is outright hostile to those skills - it's very difficult for them to compete in the jobs market.
Poor white girls tend to develop more of those soft skills. There are lots of poor white girls helping taking care of siblings, family members, putting the tea on, chatting with gran etc - while boys are much less likely to be given similar chores or be as involved in the "chatting" of a family.
Their interactions with male friends involve gaming, physical activities etc - again, less likely to have social skills of girls from the same background.
Poor white women have higher employment levels than men so they also much more likely see their fathers/uncles/older brothers being unemployed while female relatives are more likely to work part or full time. Following dad into a trade is less likely than a girl following her mum into retail work/care work etc.
Work for people with no qualifications in some areas is much more plentiful in "female jobs" like retail and care work than very physical jobs that men have an advantage in. So the work that is available to them is "women's work". Why bother to learn anything at school if you don't see a path ahead?
I think there is often a reluctance of the fathers to push their sons to do better at school in part because this would essentially acknowledge that they themselves could have done better/earned more/been happier if they were better educated. That's a hard blow for their pride.
Immigrant parents are much more comfortable with the idea that their lives would have been better with better education - hence they are much more comfortable to push their kids.