Will WHOIS requests leak my address?

Nope. Even though you must supply your address in the registration form, a WHOIS request for your locality domain will only show information about the registrar.

Perhaps I am misunderstanding their statement but unless something recently changed this is not true. The .US TLD does not permit whois privacy services. The full legal name and address of the registrant will be shown in my experience and I could not find a registrar that would deviate from this.

Are they offering delegation of sub-domains of some domains they purchased perhaps? The example they gave did not suggest this if that is so. If that is the case then whois does not really apply unless they are giving different answers in their whois for sub-domains assuming their whois would be queried.

That is why I opted for .org for a small town that I operated a website for in my spare time. When using a .US one can register it in the name of a company or the city can register it themselves through their own government to avoid a persons personal information being listed. Ensure auto-renew is enabled when assisting a city government as people come and go. Pay as far in advance for as many years as possible.