The price point may not be ideal but Business plan websites can set up branded subdomains (eg ns1.example.com, ns2.example.com).
Also, there are a limited amount of nameservers (2550 possible combinations with 50 male and 51 female nameservers I believe), so there’s probably a few thousand other accounts with the same combination of nameservers as you. This can still be a problem though, if you use a less-common TLD and/or you use pages with similar content.
You could try using the API to automate this across multiple accounts https://api.cloudflare.com.
No idea how I missed custom nameservers on business plan.. Hope it's something they added this year, otherwise I feel quite embarrassed.. It's too pricey for personal use (since the plans are associated with websites and not accounts) but the company will probably upgrade a few of the websites, thanks!
Also, it's currently only possible to avoid leaking that multiple websites belong to the same entity by creating multiple Cloudflare accounts due to the nameservers. This means that if we want to use Cloudflare workers for our 5 unique nameservers, then we would have to do a lot of unnecessary duplication. So on a somewhat unrelated note, do you have any plans to add support for adding additional nameservers to a Cloudflare account (this is currently holding us back from using Workers over Lambda, and it's something we would gladly pay for).