The first major VMware version which does not support the installable desktop client has a plenty good enough HTML5 version of the web client. So either/or, you should be good to control your hypervisors without Flash.
And at worst, you shouldn't need to use a Flash-based UI to upgrade VCSA or your hypervisors.
That's false. The old thick client stopped working with 6.0, which got a very poor, limited and extremely buggy HTML5 client in one of its latest updates. 6.5, latest update, has a somewhat usable but still limited ( missing features) HTML5 UI ( funnily enough some new features were only in the HTML5 UI, while some others were only in the Flash version). The first version that had a fully usable for everything HTML5 UI is 6.7 Update 3, which came out a few months ago.