Registering a New Domain on Different Hosting W/Private WhoIs... Question About Virtual CC

So I want to register a new domain to use in conjunction with another site on a similar niche but rather than adding it to my existing hosting and all of that I thought I'd try to give it a unique appearance as if it has no affiliation with my initial site.

I'm going to use NameCheap for the first time because I understand you get free private registration with all new domains, and I'll use HostGator hosting (separate from my GoDaddy hosting for my first domain). Should I also go to the trouble of getting a virtual credit card and paying with that as opposed to using my own credit card or Paypal? I heard someone argue that if you use a credit card or account tied to your name, someone could still find out that you're behind that site, and the only way to be truly anonymous was to use a virtual CC.

Is there any truth to this/is this recommended? I've never used a virtual CC and know nothing about what is preferable with what I'm trying to do. Thanks for the help.

Google and others can only track you up to your registration information in your whois. If they have access to how you've paid for your domain, there would be some serious security issues there.

Okay so it really is a non issue then, not sure why someone suggested that extra step. Thank you.