I'm really trying to remove any of the date information, and get to a custom version string that isn't just [Major.Minor]-[Status]... Ideally something like [token1].[token2].[token3], with a reserved shortcut for just [token] (no major/minor/status stuff at all).

I think I've gotten the equals/hashcode stuff worked out, I'm just getting stuck on how to wire in the Provider/Formatter/Reader... the codepath bounces between all of them, and its tough to debug through to understand what's going on as there are so many partial classes :-/

Are there any examples of a successful custom version format?

It's not terribly complex, but there are a lot of moving parts. I don't know of anyone that has bothered to put in the full effort required to actually implement a custom API version. It's a bit dense for here so I created the following Gist:

https://gist.github.com/commonsensesoftware/e7280cc49381928b2297f1f4e5314c36

Once implemented, the setup would change to something like:

services.AddSingleton<IApiVersionParser>( CustomApiVersionParser.Default );
services.AddApiVersioning( options =>
{
    options.DefaultApiVersion = new CustomApiVersion( "*", "*" );
    options.ReportApiVersions = true;
} );

For a Minimal API, the configuration might be:

var forecast = app.NewVersionedApi();

// GET /weatherforecast?api-version=A.1
forecast.MapGet( "/weatherforecast", () => Results.Ok() )
        .HasApiVersion( new CustomApiVersion( "A", "1" ) ); // ← built-in extension methods cannot be used here; create new ones

For a controller-based API, it would be more like:

[ApiVersion("A.1")]     // ← parses using CustomApiVersionParser.Default
[ApiVersion("B", "2")]  // ← no parsing required
[ApiController]
[Route("[controller]")]
public class WeatherForecastController : ControllerBase
{
    [HttpGet]
    public IActionResult Get() => Ok();
}

I tested this out and if the built-in ApiVersion is used, it no longer matches. Similarly, using ?api-version=A.1 or ?api-version=B.2, matches as expected. You shouldn't need any custom readers. Formatting is optional and primarily useful for OpenAPI. I've given you a partial implementation, but you'd have to define your own format codes and how they are implemented. I hope that helps get you started.