Register async factory method with parameters in ASP.NET Core DI container

Question

I have the following (rather simplified) code:

public class Service
{
    private IPage? _page;

    private readonly ScreenshotOptions _options;

    public Service(IOptions<ScreenshotOptions> options) => _options = options.Value;

    public async Task CreateScreenshotAsync()
    {
        if (_page == null)
            _page = await InitializePlaywrightAsync();

        await _page.ScreenshotAsync(new PageScreenshotOptions { Path = _options.Filename });

        // all the other stuff
    }

    private async Task<IPage> InitializePlaywrightAsync()
    {
        var playwright = await Playwright.CreateAsync();
        var browser = await playwright.Chromium.LaunchAsync();
        var page = await browser.NewPageAsync();
        await page.SetViewportSizeAsync(_options.Width, _options.Height);

        return page;
    }
}

// Program.cs
builder.Services.AddSingleton<Service>();

I'd prefer to have InitializePlaywrightAsync() not inside Service but let the ASP.NET Core DI container do the work of instantiating IPage. Unfortunately, the following code does not compile:

public class Service
{
    private IPage _page;

    private readonly ScreenshotOptions _options;

    public Service(IOptions<ScreenshotOptions> options, IPage page)
    {
        _options = options.Value;
        _page = page;
    }

    public async Task CreateScreenshotAsync()
    {
        await _page.ScreenshotAsync(new PageScreenshotOptions { Path = _options.Filename });

        // all the other stuff
    }
}

// Program.cs - compilation error
builder.Services.AddSingleton<IPage>((IOptions<ScreenshotOptions> options) => 
{
    var playwright = await Playwright.CreateAsync();
    var browser = await playwright.Chromium.LaunchAsync();
    var page = await browser.NewPageAsync();
    await page.SetViewportSizeAsync(options.Width, options.Height);

    return page;
});
builder.Services.AddSingleton<Service>();

Using a dedicated factory class also is not super nice as the DI container does not accept an asynchronous delegate:

public class PageFactory
{
    private readonly ScreenshotOptions _options;

    public PageFactory(IOptions<ScreenshotOptions> options) => _options = options.Value;

    public async Task<IPage> InitializePlaywrightAsync()
    {
        var playwright = await Playwright.CreateAsync();
        var browser = await playwright.Chromium.LaunchAsync();
        var page = await browser.NewPageAsync();
        await page.SetViewportSizeAsync(_options.Width, _options.Height);

        return page;
    }
}

// Program.cs - compilation error
builder.Services.AddSingleton<IPage>(async provider => await provider.GetRequiredService<PageFactory>().InitializePlaywrightAsync());

I'm curious whether there is any elegant way of moving the InitializePlaywrightAsync() logic into the DI container.