aspNetHidden div not being served depending on client

In short, it is not documented/specified in terms of useragents, but browser capabilities.

Based on the browsers useragent a set of capabilities gets set up.
These capabilities are configured in .browser configuration files on the webserver.
For e.g. .NET 4 you find these files in %SystemRoot%\Microsoft.NET\Framework\v4.0.30319\config\browsers,
e.g. chrome.browser, iphone.browser, etc.

Such a .browser file contains a tagwriter capability.
E.g. chrome.browser:

<browsers>
    <!-- Mozilla/5.0 (Windows; U; Windows NT 6.0; en-US) AppleWebKit/530.1 (KHTML, like Gecko) Chrome/2.0.168.0 Safari/530.1 -->
    <browser id="Chrome" parentID="WebKit">
        <identification>
            <userAgent match="Chrome/(?'version'(?'major'\d+)(\.(?'minor'\d+)?)\w*)" />
        </identification>

        <capabilities>
          <capability name="browser"   value="Chrome" />
          <capability name="tagwriter" value="System.Web.UI.HtmlTextWriter" />

          <!-- ... -->  
        </capabilities>
    </browser>
</browsers> 

The tagwriter capability specifies whether a System.Web.UI.HtmlTextWriter or a System.Web.UI.Html32TextWriter will be be instantiated to write the output.

The default configuration in the Default.browser file, declares tagwriter as:

<capability name="tagwriter" value="System.Web.UI.Html32TextWriter" />

Also, if the tagwriter capability is missing a Html32TextWriter is being used.
From the Microsoft reference source:

internal HtmlTextWriter CreateHtmlTextWriterInternal(TextWriter tw) {
    Type tagWriter = TagWriter;
    if (tagWriter != null) {
        return Page.CreateHtmlTextWriterFromType(tw, tagWriter);
    }

    // Fall back to Html 3.2
    return new Html32TextWriter(tw);
}

The Html32TextWriter declares not to render a div around hidden input fields.
From the Microsoft reference source:

internal override bool RenderDivAroundHiddenInputs {
    get {
        return false;
    }
}

The HtmlTextWriter does return true for RenderDivAroundHiddenInputs, see the Microsoft reference source.

Some more reading about all this here.

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What you can do.

If you always want the wrapping div, use one of the wellknown useragents, otherwise use a custom one like the Test Client you are already using.
If you control the website being requested, you can set up a custom .browser file for your custom useragent ... but I would rather not go that way ...

When making the request, just set the appropriate User-Agent request header on your HttpClient, e.g.:

var client = new HttpClient();
var userAgent = "Test Client"; // Or "Mozilla/5.0 (Windows NT 10.0; Win64; x64) AppleWebKit/537.36 (KHTML, like Gecko) Chrome/83.0.4103.61 Safari/537.36 Edg/83.0.478.37"
client.DefaultRequestHeaders.Add("User-Agent", userAgent);