People seem to misunderstand NHS/mixed dentistry in a way they don't seem to with pharmacy or optometery, which also operate on mixed NHS/private models.
For example, say you have a free prescription for Ibuprofen. You can take it to your local pharmacy with an NHS contract and they will give you the standard-issue generic ibuprofen, provided they have enough. If they don't, you can either try another pharmacy, or 'go private' and buy Nurofen or something for £5.
Because the NHS don't fund the Nurofen, the pharmacist wouldn't be reimbursed for it and isn't just going to give it to you for free, so you'd have to pay for it.
Likewise, if you took your prescription to the corner shop and tried to use it to get a free packet of ibuprofen, they wouldn't accept it and would have to sell you a pack instead.
And if there was a national shortage of generic ibuprofen then you'd either have to go without or suck it up and pay for branded versions.
If you replace 'prescription' with 'NHS dentistry' and 'ibuprofen' with 'treatment', that's basically how NHS dentistry works. And because there is currently a supply problem - for all the reasons outlined above (fallow time, PPE donning/doffing, extra cleaning, backlog of patients, etc, etc) - dentists are offering to see people privately (i.e. letting people 'buy the Nurofen') rather than make them go without or have to search high and low for another practice with space.