Note that this is only true when not using eco mode, which opens the door right away so you can just chuck the tablet in like a caveman. Mine happens to work well enough in eco mode and the little door tends to get stuck on things, so that's what I usually do.
If your (modern) dishwasher isn't cleaning properly in eco mode, either you have very hard water, or you should clean the filter more often.
If you don't remember when the last time you cleaned the filter is, go clean it now.