It looks like the only reason that InitClientTimings expects a WebBasePage is because it is an extension method. It seems like this limits the ClientTimings to only Razor based pages and excludes WebForms. Are there other implications to this or if we add a WebForm (System.Web.UI.Control) implementation would that be enough to support client timings in WebForms?
This would be a bug then, changing it to one. There should be no arbitrary restrictions here for web forms