Why doesn't codec tweak tool have taskbar entry?

There is no way to know there are updates because update checker window always gets covered by other windows.
Also why can't installer close Codectweaktool.exe by itself?

It shows in background because people don't want it to interfere with their work/gaming/etc. You should see it after closing windows.

The installer is capable of terminating the exe. Isn't that working for you? It first asks to close it yourself.

Is it possible to disable hiding updater from taskbar? It is OS task to not disturb you when you in a fullscreen application and I'm sure that it will not pop up window in that case. I always see update notification when I going to shutdown my computer and of course I cannot wait while site will load content with actual download links which is not included in the initial page markup and dynamically inserted (or not) into page after several minutes of waiting.

Please provide an option to enable taskbar button. Some registry switch. Anything. Thanks.

The download link is directly visible on the download page.

How slow is your internet?