If I want to enable statistics for Web Services subsystem, I do it in CLI:

[standalone@localhost:9990 /] /subsystem=webservices:write-attribute(name=statistics-enabled, value=true)
{"outcome" => "success"}

Via web console, I navigate to Configuration->Subsystems->Web Services->View
I click edit and click checkbox for statistics enabled attribute. Now when I submit the value by clicking Save, dialog appears with reload require information.

There is no need to require reload just for enabling statistics, by the way statistics are accessible even before reload (Runtime->Subsystems->Webservices and try to get wsdl for the web service for several times).

Note: if there is no active deployment, reload is not required in web console (correct). To reproduce the issue you must have some active WS deployment