After seeing that video my money is on a bad servo. Its like its not able to hold center and you mention in the video that it doesn't have enough power to turn the wheels properly. That suggests bad servo also.
I agree, I would upgrade to a 2.4Ghz radio first and see if that solves the problem. You'll like the range and features better than an AM TX/RX anyway even if its just a budget radio like the Flysky GT3b or GT3c.
If the radio doesn't fix it then the servo would be my next suspect