If someone works 100% remotely, why should their salary be tied to which city they are in? They do exactly the same job if they are in a big city or village.

Actually all major IT companies told exactly this since going to remote work - if you go into cheaper state, your salary will be cut. If you won't inform company and will try to keep getting same money - you will be fired.

In a regime of big industry or of free competition—as we shall see, the two come to the same thing—the price of a commodity is, on the average, always equal to its cost of production. Hence, the price of labor is also equal to the cost of production of labor. But, the costs of production of labor consist of precisely the quantity of means of subsistence necessary to enable the worker to continue working. The worker will therefore get no more for his labor than is necessary for this purpose; the price of labor, or the wage, will, in other words, be the lowest, the minimum, required for the maintenance of life (at the standard usual for specified place).

In cheaper state your labor price is lower, so capitalist do not want to pay you previous amount of money.

Btw it also explain why modern capitalist also badly want you to not have children, as lack of children expenses also reduce labor costs. And present capitalist does not care about the future situation with lack of new labor.