No response to formal Chair application. Is this normal? I have applied to a Chair position, in response to a public advertisement from a US University (not one of the top ones, of course). This included investing some good time, sending several documents, and a formal application (the first, and probably last one).

In 2 years, I never heard 1 word in response (apart the application reception acknowledgment). I was wondering if this kind of rudeness is the standard in the USA, and it is to be normally expected.
This is the typical situation normal but all fouled up (SNAFU) of job searches in the United States. Often you are lucky to get a two line e-mail or a postcard at the negative conclusion of a search. 

In the case of Chair searches, though, the fouled-up nature is usually much worse. I'm assuming this is Chair-as-in-Head-of-a-Department rather than a Named-Chair-as-in-Benefactor-Sponsored senior position. 

Usually universities search for external Chairs when there are problems with the existing senior faculty that make it impossible for a Chair to arise from within the department. This means that search will be similarly confused, with possible internal conflicts in the department as well as external interventions from the Provost's office.

Because of these competing interests, external Chair searches can take even longer than the usual search. 

I would write to the Search Chair and ask what the status of the search is. 

Postscript: The OP has clarified that this is a search for a named chair. These tend to go smoothly as the institution already has the money vested by the benefactor, who usually wants to see the position filled quickly (and is likely to give again if pleased by the results) and often puts enough parameters on the position (theoretical field, area of study) that political factions within the department can't mess it up too much. So whatever is going on in the OP's situation seems strange and rather unique.