The crux of your question is whether you as a customer can set the conditions for a business to operate. So long as the business is not breaking laws (and unfortunately even sometimes if they are) you cannot do more than complain to authorities that due to existing legislation, resources, and priorities are often disinterested beyond some arm waving, to be polite…
There is always some risk of identity theft in any transaction involving personal details. One has to weigh up the probability not just the possibility.
Many countries, Australia seemingly at the fore, tries to legislate to an Nth degree from DIY activities to having >512 pages of road rules in most states when <100 do it in many jurisdictions.
Have you ever given your credit card details to someone unknown to you by phone? It was and remains common when booking accommodation when not using a booking platform (that have their own security and financial risks). They will have your address, name, and card details. There are regular data breaches from web sites.
Thus my personal answer to your question is ‘no’. The ‘new requirement’ if the DL was restricted could be posting a bond. Would/could you post say a $500 cash or credit card bond for a golf buggy when they might rightly or wrongly claim damages and try to keep it? Another choice to consider. Also consider the process for hire cars.
