Agree it could help, higher productivity leads to a higher tax take, but as you say, based on history, its not clear the UK is capable of achieving this, any more than any other major European country.
So, Opinion(!); The UK economy is stuck in a low growth, high tax, high regulation, low productivity recurring cycle. The political system offers voters rewards that the country simply cannot afford but no one is prepared to admit it because the whole facade would fall down. (All Parties!). We need serious root cause analysis with a long term plan based on an honest assessment of the situation, and instead we see the blame game, short term imaginary quick fixes and grand projects; and the re-enabling of the BOE magic money tree when the next recession hits (just dont call it QE!).
Either way there's no serious plan anywhere to address gov debt. Sure, the UK will not default, cos it still prints its own money. But the cost of that debt to ordinary people (eg inflation; decline of home ownership; lack of financial independence for young people, etc) will increase, along with social problems/division continuing to worsen...
Course, I could be wrong...