The diagram above has a very important feature.
It is the connection between the Earth and the outer conducting shell.
Assume that the Earth is a conducting sphere and has some net positive charge on it.
This will mean that the outer shell connected to it will also have some positive charge on it but the wire between the outer shell and the Earth means that they are at the same potential.
Now what is the value of that potential?
You can call it zero and then infinity will be at a negative potential or you could have infinity at zero potential and then the outer shell and the Earth would be at the same positive potential.
Starting from the beginning with no Earth connection.
For simplicity let the zero of potential be infinity and the Earth and the inner and outer shells having no net charge on them also having a potential of zero.
If the Earth does have a charge in what follows the only change is that the potentials all change by an amount equal to the initial (non-zero) potential of the Earth.
Move a charge of $-Q$ from the Earth to the inner conducting shell which leaves a net charge of $+Q$ on the Earth.
Charges on the inside of the outer shell will redistribute themselves as in the diagram and there will be a net charge of $-Q$ on the outside of the outer shell.
That negative charge will not reside evenly on the outside of the outer shell rather there will be more near the Earth as there will also be more positive charge on the Earth in that region.
In terms of potential relative to infinity the potential of the outer shell has decreased and that of the Earth has increased but not by the same amount.
Because the Earth is so big relative to the outer shell the redistribution of the charge on the Earth is greatest near the outer shell and negligible on the other side of a diameter.
So the potential of the Earth relative to infinity has increased hardly at all.
What you cannot do is assume that the Earth is an isolated sphere and apply the $+Q=C_{\text{Earth}} \Delta V$ formula to find the change in potential of the Earth.
The Earth is not isolated, it is under the influence of the negatively charged outer shell and this in turn means that charges are not uniformly distributed on the surface of the outer shell or the Earth.
Now connect the negatively charged outer shell to the positively charged Earth.
Charges will flow until the potential difference between the outer shell and the Earth is zero when the net charge on the outside of the outer shell and the Earth is zero.
The potential of the outer shell and the Earth will be zero.
The redistribution of charge occurs locally and the area of the local region is very, very much smaller than the surface area of the earth.
So a 10 F capacitor with one terminal earthed when charged with $+1$ coulomb will leave a charge on the Earth of $-1$ coulomb close to it thus disturbing the local potentials but the potential of the Earth as a whole hardly at all.
I did want to try and make the answer more quantitative but this interesting article “Electrostatics of two charged conducting spheres” made me realise that finding and then applying the capacitance of two unequal sized spheres is not a trivial matter.