Tested quickly under 10.6.8. It appears to work.
1. Copy (don't move) QuickTime Player.app to the Desktop.
2. Modify the copied app. Control click its icon. Select Show Package Contents.
A. Navigate to
/Contents/Resources/English.lproj/Localizable.strings
Open this file with a text editor such as TextWrangler. Change this line:
"Stop Recording" = "Stop Recording";
to this:
"Stop Recording" = "";
That modification removes the text from the menubar.
B. Next, the small black circle. It's located in
/Contents/Resources/RecordingControls_Stop.png
Move RecordingControls_Stop.png to the trash.
3. Test the modified app.
Notes:
--To stop a screen recording, use Cmd-Cntl-Esc or click the menubar where the Stop Recording icon used to be. (Even though there's nothing visible in the menubar, there's a vestigial reserved space that continues to function as a stop recording command.)
--There's only one RecordingControls_Stop.png for the app. It's used (at least) in both the menubar and the dialog that appears as you begin a screen recording. That black circle will no longer appear anywhere as you use the app.
--You can of course move the modified QuickTime Player.app off the Desktop to another location. If you want to put it in the Applications folder, rename it as "QuickTime Player-zirpoli.app" (or something along those lines) to keep it from overwriting the original QuickTime Player.app. The original QuickTime Player.app should be left undisturbed in the Applications folder to avoid causing future problems.