Here is a really good explanation of the different controller types

https://www.jondjones.com/learn-umbraco-cms/umbraco-9-tutorials/getting-started-with-umbraco-9/ultimate-umbraco-v9-controller-guide/

I tried following that guide and based on the description of the controller types I think I should be using a vanilla MVC controller. So along with the umbraco documentation I've tried creating a custom controller.

Here's my ViewModel:

public class CountyBirthViewModel : ContentModel
{
    public CountyBirthViewModel(IPublishedContent content)
        : base(content)
    {
    }

    public string FirstName { get; set; } = string.Empty;
    public bool HasSearched { get; set; } = false;
    public IEnumerable<CountyBirthIndex> SearchResults { get; set; } = Enumerable.Empty<CountyBirthIndex>();
}

Controller:

usings...

namespace Umbraco_Prod.Controllers
{
public class CountyBirthController : UmbracoPageController, IVirtualPageController
{
    private readonly Library_Context _libraryContext;
    private readonly IPublishedValueFallback _publishedValueFallback;
    private readonly IUmbracoContextAccessor _umbracoContextAccessor;

    public CountyBirthController(ILogger<CountyBirthController> logger, ICompositeViewEngine compositeViewEngine, IUmbracoContextAccessor umbracoContextAccessor, IPublishedValueFallback publishedValueFallback, DplLibrary_Context libraryContext)
        : base(logger, compositeViewEngine)
    {
        _libraryContext = libraryContext;
        _umbracoContextAccessor = umbracoContextAccessor;
        _publishedValueFallback = publishedValueFallback;
    }

    public IPublishedContent FindContent(ActionExecutingContext actionExecutingContext)
    {
        if (_umbracoContextAccessor.TryGetUmbracoContext(out var umbracoContext))
        {
            var countyBirthPage = umbracoContext.Content.GetById(19708);
            if (countyBirthPage != null)
            {

                if (actionExecutingContext.ActionDescriptor is ControllerActionDescriptor controllerActionDescriptor)
                {
                    // Check which action is executing
                    switch (controllerActionDescriptor.ActionName)
                    {
                        case nameof(Index):
                            return countyBirthPage;
                    }
                }

                return countyBirthPage;
            }
        }

        return null;
    }

    [HttpGet]
    public IActionResult Index()
    {
        return View(CurrentPage);
    }

And here's the beginning of my view:

@using Umbraco.Cms.Web.Common.PublishedModels;
@inherits Umbraco.Cms.Web.Common.Views.UmbracoViewPage<Umbraco_Prod.Models.DPL.CountyBirthViewModel>
@{
    Layout = "Masterpage.cshtml";
}

...header html

        @*Search Form*@
        <div class="row">
            <div class="col-md-6">
                <form asp-action="index" method="post">
                    <div class="form-group">
                        <label asp-for="FirstName" class="control-label">First Name</label>
                        <input asp-for="FirstName" class="form-control" />
                    </div>
                    <div class="form-group mt-3">
                        <input type="submit" value="Search" class="btn btn-primary" />
                    </div>
                </form>
            </div>
        </div>

Finally, here's the error I get from the Index action, "ModelBindingException: Cannot bind source type Umbraco.Cms.Web.Common.PublishedModels.CountyBirth to model type Umbraco_Prod.Models.CountyBirthViewModel. The source is a ModelsBuilder type, but the view model is not. The application is in an unstable state and should be restarted."

Is there something wrong with my using and inherits statements in the view? Should I be doing this in a render controller instead of vanilla?

Could be your viewmodel, it should use CountyBirthViewModel : CountyBirth rather than CountyBirthViewModel : ContentModel

I've got this working using a RenderController and passing all the form values in the querystring, but that just feels like a sloppy way to go about retrieving and displaying data from an external source. I was hoping I'd be able to do this using a strongly typed form with data binding to the viewmodel, but for that it seems I need to do some sort of mix between render controller for the doc type/template route hijacking and a surface controller for the form; but even if I go that route all the examples I found still pass the form values back to the render controller in the query string. The more I look at this the more it seems like I should have gone with an UmbracoApiController, posted the form through ajax, and build the table from the response.