You say the shelf is sealed ... I've been to quite a few that are sealed with silicone all round. Some builders will often do this and seal everything, because all the customer sees is a shiney new bathroom. They dont notice that their are no access panels...and by the time anything goes wrong the builder will be long gone.
( I'm not saying ..'The Builder'.. I am sure he's marvelous) just the builder who fitted the bathroom.
Usually I cut the sealant away around the shelf with a stanley knife,remove the shelf,and gain access through the top of the cistern... I went to one a few weeks ago where I had to cut the tiles out above the pan along with the plasterboard to create an access panel. Repairing the cistern before, reparing the wall and then re-tiling.
It's quite common...
If water is entering the cistern, it may well be your float valve not shutting off. Or you may have a flush valve leaking which would cause the cistern to keep filling.
It really depends on what setup you have.
Keyhole surgery ... and working through the button would be a new one on me..!