Possibly you're taking the wrong counterfactual here?
"If I hadn't HTB but everyone else did" vs "If no-one had HTB".
It's possible you're quite right about the first, but it tells us little about the second. Possibly in the presence of HTB, prices just rise again until they reach their equilibrium of just-barely-affordable.
If they do pay it back they do it by remortgaging so the builders win, the banks win, the tax payer pays for any HTB loans that aren’t repaid and the tax payer gets shafted with higher housing costs.
For a £500K flat you need a £50K deposit but you also need a household income of £100K or more to be able to borrow £450K as banks cap the maximum amount they’ll loans at 4-4.5 of your annual income.
So HTB didn’t only increase the prices it also pushed people to take on loans that they normally would not be able to afford.