Sports injuries are accidents, not quite the same.
Dental disease is more like something caused by smoking, alcohol or poor diet, or obesity... Say, heart disease, lung disease etc. The difference is that no one's really going to die if they can't afford dental treatment. If the NHS put a price on treatment that is entirely 'self inflicted', people would drop like flies, they wouldn't be able to afford it. There would be outrage. Imagine someone rocks up needing a bypass due to 50 years of wilful neglect of their body and is told... You'll need to pay £10,000 for your operation. Most of the population could not pay that. But someone comes in to me needing a filling, we charge them a fiver. Big difference.
Dental disease is not (generally) life threatening in the same way systemic diseases are. For most people it can be prevented entirely by brushing your teeth and consuming less sugar. The NHS pays 20% of all your already low costs. I think it's a fair deal, personally. I'd rather pay a small amount for my dental work than covering costs of someone else who wilfully neglects their oral health through taxes. The cost to the NHS if they covered everything would be unthinkable.
It does, however, infuriate me that I have to pay privately for my medication for autoimmune disease (£32 pm + doctors fees!) when those with addictions to drugs/alcohol get it in far higher doses for free...