Tank cannot be pressurized from just filling from the sink and tub drains. It would just fill until the tub backs up and overflows. No pressure it is just gravity filled and vented. Even if vent was clogged it would still fill to a certain point and them back up into the tub and overflow. At most there could only be about a half pound of pressure in the tank if the vent pipe becomes clogged due to a foot to 18 inches of head (pressure due to weight of water) above the tank. And this would be normal for a tank that has a closed valve that was full with water backed up into the tub.
So they are full of it when they tell you the leak is because the tank is poressureized.
There are 2 connection methods for the vent and fill connections on top of tank.
One is female threaded fittings in the top of the tank with male fittings screws into them and the vent and fill pipes are glued to the male adapter fitting.
The second connection method are holes cut in the top of the tank and a rubber grommet inserted into the holes, then the vent and fill pipes are pushed into the grommets which should become tight when the pipe is inserted.
Potential problem is one of the following:
Fill pipe not connected.
No gasket or wrong fittings at fill pipe connection.
Vent pipe not connected.
No gasket or wrong fittings at vent pipe connection.
"New" Grey Tank is cracked.
Tank was never replaced and they tried to patch it.
They did not loosen the vent pipe at the roof so they could drop it down so they could properly connect it to the new tank. Then reseal the pipe at the roof when the new tank is pushed back up into place along with the pipe. Did they go up on the roof?
The same would need to happen with the tank fill pipe. Did they go into the cabinets or plumbing access for the tub and sink drain piping to disconnect that pipe above the floor?