There are issues with the webpart specifically when there is some js code that conflicts or has errors. By design the webpart does not execute the javascript when in the design or edit mode as to not conflict with the UI pieces. What is most likely happening is either you have some conflicting JS or bad HTML.
In the event you are unable to use the Design tab to remove the webpart you have 2 options:
- Go to the Templates app, find the template, on the design tab, attempt to remove the webpart there.
- If #1 does not work, then in the Templates app on the template in question, go to the Webparts tab and look through the raw XML and find the
<webpart>
node and remove it.
If none of these things resolve it, I'm assuming you tried the items in the accepted answer on this post you already commented on?
Lastly, if people would provide more information when asking questions then we wouldn't have to ask more questions. Instead of criticizing the MVPs for asking questions, you should be thankful they are willing to take time out of their day to help others (with poorly formed questions) with their issues. If this question was on Stackoverflow, it would have been down voted and flagged for removal because of the lack of information.
P.S. if you don't like the answers (or questions) you get from volunteers on the DevNet, I suggest contacting Kentico Support, where they too, will ask you questions.