Just out of interest, who would you guys usually expect to pay at a birthday meal?
It's completely irrelevant who I or anyone else would expect to pay for anything. The important thing, especially when organising any with so many people who do not have reliably shared cultural experiences is to be completely upfront and clear. You can best do that by setting out yourself how it's going to work right up front and remind everyone in every communication.
The restaurant are confident they don't need a deposit because they have you as their contact.
Now is certainly the time to tell the restaurant that everyone will be responsible for their own bill for good and drinks.
Snd update your invitation saying something g like 'in response to the many questions, the restaurant are aware that everyone will be responsible for their bill for anything they order. Please ensure you make clear to the staff that you will be billed for the things you have ordered.
And be aware that you do not need to play host by getting anything for anyone, or you will end up paying for them.
Ad it happens, in my culture, once adult, the birthday child invited and picks up the bill.
I know this is not the same everywhere and am very happy to go with whatever the organiser wants to do. The last thing snyone wants is the potential embarrassment that comes from a lack of clarity