I feel guilty for not doing it >.>. I did it on single player if you want, it turned out to have a very simple and compact solution which I hadn't thought of before doing some tests :3.
Basically, the button will dispense any block, in this case dirt, onto a wooden pressure plate (if you are going to pipe in your trash into this system, be careful to use a dropper, just in case you throw away arrows/snowballs or other things that will be shot out instead of dropped as items from a dispenser). This plate will turn the power on for the minecart dispenser. Since the wooden pressure plate is adjacent to the trash dispenser, this will also power the trash dispenser. When a dispenser is constantly powered, it will dispense once and once only regardless of if you power it again, so until the item despawns, pressing the button won't do anything, even waste more trash

. The system will be locked until 5 minutes after the user has pressed the button and the item is gone.
Note that this is 5 minutes of the chunk being loaded, which is going to be significantly longer in most cases than 5 minutes. I've included some screenshots and a
world download :3.
To use the world download, unzip the file and put the folder in C:\Users\YOURUSERNAMEHERE\AppData\Roaming\.minecraft\saves
Screenshots below:
Simple Version
Fanceh Version
