The answer turns out to be caused by more of Microsoft's Broken by default policy.
Rather than acting as a web-server, accepting requests and handling them, ASP.net by default decides to ignore the majority of requests - because it thinks the user shouldn't be doing them.
The solution is to rip everything related to ASP.net out of IIS, and then re-add it correctly:
web.config
<?xml version="1.0"?>
<configuration>
<modules runAllManagedModulesForAllRequests="true">
<!--IISFIX: Whatever this is, it causes 405 Method Not Allowed errors on IIS when using PUT. (Microsoft's broken by defult)-->
<remove name="WebDAVModule"/>
</modules>
<handlers>
<!--IISFIX: ASP.net is broken by default. By default they will not accept verbs from the client.
First we have to rip out everything related to ASP.net-->
<remove name="ExtensionlessUrlHandler-ISAPI-4.0_64bit"/>
<remove name="ExtensionlessUrlHandler-Integrated-4.0"/>
<remove name="ExtensionlessUrlHandler-ISAPI-4.0_32bit"/>
<remove name="SimpleHandlerFactory-ISAPI-2.0-64"/>
<remove name="SimpleHandlerFactory-ISAPI-2.0"/>
<remove name="SimpleHandlerFactory-Integrated"/>
<remove name="SimpleHandlerFactory-Integrated-4.0"/>
<remove name="SimpleHandlerFactory-ISAPI-4.0_64bit"/>
<remove name="SimpleHandlerFactory-ISAPI-4.0_32bit"/>
<!-- IISFIX: Now that we're ripped out everything related to ASP.net, put them back correctly.-->
<add name="SimpleHandlerFactory-ISAPI-4.0_32bit" path="*.ashx" verb="*" modules="IsapiModule" scriptProcessor="C:\Windows\Microsoft.NET\Framework\v4.0.30319\aspnet_isapi.dll" resourceType="Unspecified" requireAccess="Script" preCondition="classicMode,runtimeVersionv4.0,bitness32" responseBufferLimit="0"/>
<add name="SimpleHandlerFactory-ISAPI-4.0_64bit" path="*.ashx" verb="*" modules="IsapiModule" scriptProcessor="C:\Windows\Microsoft.NET\Framework64\v4.0.30319\aspnet_isapi.dll" resourceType="Unspecified" requireAccess="Script" preCondition="classicMode,runtimeVersionv4.0,bitness64" responseBufferLimit="0"/>
<add name="SimpleHandlerFactory-Integrated-4.0" path="*.ashx" verb="*" type="System.Web.UI.SimpleHandlerFactory" resourceType="Unspecified" requireAccess="Script" preCondition="integratedMode,runtimeVersionv4.0"/>
<add name="SimpleHandlerFactory-Integrated" path="*.ashx" verb="*" type="System.Web.UI.SimpleHandlerFactory" resourceType="Unspecified" requireAccess="Script" preCondition="integratedMode"/>
<add name="SimpleHandlerFactory-ISAPI-2.0" path="*.ashx" verb="*" modules="IsapiModule" scriptProcessor="%windir%\Microsoft.NET\Framework\v2.0.50727\aspnet_isapi.dll" resourceType="Unspecified" requireAccess="Script" preCondition="classicMode,runtimeVersionv2.0,bitness32" responseBufferLimit="0"/>
<add name="SimpleHandlerFactory-ISAPI-2.0-64" path="*.ashx" verb="*" modules="IsapiModule" scriptProcessor="%windir%\Microsoft.NET\Framework64\v2.0.50727\aspnet_isapi.dll" resourceType="Unspecified" requireAccess="Script" preCondition="classicMode,runtimeVersionv2.0,bitness64" responseBufferLimit="0"/>
<add name="ExtensionlessUrlHandler-ISAPI-4.0_32bit" path="*." verb="*" modules="IsapiModule" scriptProcessor="C:\Windows\Microsoft.NET\Framework\v4.0.30319\aspnet_isapi.dll" resourceType="Unspecified" requireAccess="Script" preCondition="classicMode,runtimeVersionv4.0,bitness32" responseBufferLimit="0"/>
<add name="ExtensionlessUrlHandler-Integrated-4.0" path="*." verb="*" type="System.Web.Handlers.TransferRequestHandler" resourceType="Unspecified" requireAccess="Script" preCondition="integratedMode,runtimeVersionv4.0"/>
<add name="ExtensionlessUrlHandler-ISAPI-4.0_64bit" path="*." verb="*" modules="IsapiModule" scriptProcessor="C:\Windows\Microsoft.NET\Framework64\v4.0.30319\aspnet_isapi.dll" resourceType="Unspecified" requireAccess="Script" preCondition="classicMode,runtimeVersionv4.0,bitness64" responseBufferLimit="0"/>
<!--IISFIX: WebDAV is also buggy, and interferes with client requests-->
<remove name="WebDAV"/>
</handlers>
</system.webServer>
</configuration>
The problem now is that the web-site will not work on anyone else's machine; there are now hard-coded paths to files in web.config.
Why, oh why, couldn't Microsoft just do things right.
For completeness
For my own reference, here are the other things that i need to add to web.config
every time because the defaults are wrong:
<system.web>
<httpRuntime/>
<!-- IISFIX: By default IIS hides errors-->
<customErrors mode="Off"/>
<!-- IISFIX: By default IIS ignores the browser's culture -->
<globalization culture="auto" uiCulture="auto"/>
</system.web>
<!-- ASP.net web-sites do not support WebPageTraceListener (only ASP.net web-applications)
So this section doesn't work; and does nothing.
But if Microsoft ever fixes IIS, we will start working automagically. -->
<system.diagnostics>
<trace>
<listeners>
<add name="WebPageTraceListener" type="System.Web.WebPageTraceListener, System.Web, Version=2.0.3600.0, Culture=neutral, PublicKeyToken=b03f5f7f11d50a3a"/>
</listeners>
</trace>
</system.diagnostics>
<system.webServer>
<!-- IISFIX: By default IIS ignores custom error pages -->
<httpErrors existingResponse="PassThrough"/>
</system.webServer>