It's a tricky one to get your head around.
If the "load" as seen by the inverter were elastic then yes, all it needs to do is produce a voltage and current will flow. But with an inelastic load (as the mains can be considered to be), controlling power flow by controlling voltage isn't a good mechanism.
If you have a variable bench power supply (which
must include current limiting), try controlling current through some bare LEDs (without current limiting resistor). You'll find that nothing happens as you increase the voltage until you reach the knee voltage, then the current shoots up with only a small increase in voltage (that's why you need current limiting or you'll just pop the LEDs.
Similarly, if you are trying to drive a set amount of power into the mains, if you just work on trying to generate a higher voltage then you'll find current control (and hence power control) very difficult. If you control the current and accept whatever voltage the inelastic mains provides then you'll find it easy.
Back to the plumbing analogy. You have a pump with a variable outlet pressure (controlled by pressure relief valve). Connect that to the mains* and you'll find there's a fairly sharp transition between all the pump flow going through the relief valve, and it all going through the mains. If you try and control the flow rate (eg to match what's going into your tank from another source) then you'll find it difficult. Furthermore, you'll have to sit there constantly altering the pressure to match as the mains varies.
If instead you use a variable flow rate pump (eg a positive displacement pump with variable speed drive), you'll find it easy to control the flow rate. The pressure will be whatever it is - that's pretty well fixed by the mains - give or take a bit to allow for local flow-related pressure drops in pipes.
* Bearing in mind that this is neither desirable nor legal - it's just for explanation purposes.
Casting my mind back <cough> decades, I vaguely recall doing lab work at uni on AC machines. Take one synchronous machine, driven by a DC motor for ease of power control (this was before cheap inverters), and sync it to the mains.
Turn up the field a bit, and without any input power to the motor, not a lot happens. Add input power (ie torque to the generator) and power flows out; treat the DC motor as a generator and pull power from it, power flows in - changing between motoring and generating occurs without altering the field current (and hence O/C volts).
With a weak field, there's less "magnetic strength" and if you apply a lot of input torque you'll break the AC machine out of sync - and the pens on the X-Y plotter go nuts
Adding field current increases the "strength" and so you can apply more torque and generate more output current before this happens.
We also had an engines lab in the basement - using an AC alternator as a load. With a fixed field current, the output power was controlled by the input torque, and hence by the engine power produced.
They also used one of the rigs as a backup generator during power cuts (I assume by just opening the main switch with no interlocks !) - with power controlled by the diesel governor altering power to keep speed constant.
But at the end of it, all you need to know is that the clever electronics inside the inverter take care of all this - it may be using voltage control at a low level, but it will have a fast feedback loop to control that based on power. It simply pushes the power out of it's mains terminals and it neither knows nor cares how much of that goes out through your meter and how much is used in the house.
The only time it cares about what's on the other side of those terminals is if it determines that it's not connected to a
solid mains. Then it shuts down as required by regs to prevent "islanding" - where a fault occurs and instead of a section of the network going dead, it carries on powered by embedded generation at an arbitrary voltage and phase relative to the main grid. Cause that and you'll have Westie and his colleagues "quite annoyed".
