We can hide/show the export option in server. we cannot disable cross tab only in view.
if you are using tableau 9.2 then yes we have also seen the issue while downloading Data or CrossTab in case if user have permission to download underlying / summary data.
This is the existing functionality in the latest version. if you have to download the data then please click on sheet somewhere or click on white / blank space on report and then try to download, the options to download Data / Crosstab will be available after that.
Please let me know if it helps !!