All home remedies whose aqueous solutions react acidically should be used with caution with cement building products (in this case: mineral grout)!
The reason for my advice is that acidic grouts destroy the (alkaline) cement stone found in all cement-based building chemicals. As a result, it inevitably also loses its strength, slowly rubs out of the joint chamber or, conversely, crumbles. There can be no "longevity" of coloring mineral grout mixes. Especially not in highly stressed components such as ceramic tile floors. It is inevitable that contaminants of all sorts will settle in the joints or on the grout. And they cannot always be dissolved by an acidic environment. Especially grease cannot be dissolved by acidic media.The problem that plagues you from an optical point of view is known from practice.
There is only one satisfactory solution, but it requires some effort: Joint renewal.
You can get the old grout out by using what is known as an oscillating saw, which is also known as an "ambulance" for cutting old plaster bandages.The new joint compound should be well-painted by the manufacturer. It may be a little more expensive than the standard cement-gray grout mix, but visually it will keep the overall impression longer. If, however, the original color has more or less changed from brown to gray, it indicates that the mixing was not done optimally at the time or that the addition of coloring pigments was kept to a minimum.