Gas plants can change load in seconds by increasing or decreasing fuel flow. You can consider the generators as operating at the local grid frequency, and power being the product of torque and frequency, so they just change torque to change load, which is done through fuel control. Aeroderivative gas turbines can go from near 0 to full load in less than 30 seconds, which is obviously an eternity compared to battery system inverters with sub 150ms settling times.
You are right that load isn’t independent of frequency, though. For those who are interested, in a simplistic and hand-wavy explanation, the torque imbalance between generation and load causes a change to the frequency. The net torque = torque of generation - torque of load = I*alpha, where alpha is the derivative of omega, or the angular frequency of the grid, and I is analogous to the inertia of the grid. If there is more generation torque than load torque on a generator (and the grid), the frequency increases and vice versa. Keeping the net torque constant, increasing the inertia makes the grid frequency derivative smaller for the same imbalance between generation and load, which is why it was typically desirable to have higher inertia synchronous generators.
What you were describing around changing fuel to maintain speed is typically frequency droop, which is where generators change their power as a function of the frequency, which is a distributed scheme for all generators to independently act to drive the torque imbalance to 0, with some insensitivity proportionality constant. For example, in California, gas turbines are assigned a droop value within the range of 3 to 5%, which means a 3 to 5 % reduction in frequency should result in a 100% increase in power, and vice versa. The total power should be provided in less than 30 sec typically.
For those that are really motivated to understand the interplay between generation, load, and frequency, look up the swing equation in the context of power system stability.
There is another aspect of synchronous generators that enable them to act to stabilize frequency independently, called the inertial response, which also has to do with their rotational energy. A generator at some frequency has KE = 0.5*J*omega^2 where J is rotational inertia and omega is angular frequency. If the frequency changes, it has a change in kinetic energy = 0.5*J*(omega1^2 - omega2^2) which is equal to some power for some period of time (= P*delta_t). This shows that as a generator sees a change in frequency, the shorter the duration, the larger the amount of energy is converted to power. Essentially, generators have an inertial response that act to inject power the faster frequency is falling, and vice versa, which is a self stabilizing function for grid frequency.
This loss of synchronous inertia as generators are replaced by inverter based resources (IBRs) is why managing grid frequency stability becomes more difficult. Various techniques are used to abate the loss of inertia, including emulating the swing equation within inverters to make them behave as synchronous generators and provide that inertial response. This is typically called grid forming with virtual synchronous machine.