As you have rightly said, Android native WebView does not "show" a scroll bar when using overflow:scroll|auto. But the scrolling is allowed when you touch and move the container around. So scroll should still work as expected, without the bouncy effect.
Another Android catch is till 4.4, the native WebView is not the one used in Chrome, i.e. ChromeView. So you are going to have to deal with it by using a javascript polyfill, like overthrow.js [http://filamentgroup.github.io/Overthrow/] or iscroll. Personally, iscroll ux sucks on Samsung Galaxy Tabs.
But if you are targetting Android 4.0>=, then this might help https://github.com/pwnall/chromeview. Haven't personally tried it though.
Check out for more details on webview changes https://developers.google.com/chrome/mobile/docs/webview/overview
Android 4.4 (KitKat) includes a new WebView component based on the Chromium open source project. ...[truncated]...., so rendering should be much more consistent between the WebView and Chrome.