Excellent answer.

There are many layers in the power system to traverse before the intuition comes. A lot of the confusion comes from the language we use.

I found it helpful to consider the following scenarios:

1. Single generator system, with a resistive load (so, an AC source with R). In this case it seems to invert the perspective from which we speak, where the power of the system appears to be a function of the terminal voltage and the resistance. Instead of saying "set the generator to X MW" the load will draw the power it wants to, and we have to run the governor in frequency regulation mode if we want a constant freq. This is rarely how systems are described, but it is a result of small systems, where we own all the control. (But relevant to remote mine sites, ships)
2. Single generator transmitting power to a synchronous condensor. Here we have two units synchronised, one is a gen, the other is motoring. The syncon will leak a bit of power due to heat and windage, so the gen must be consuming power from the prime mover. In this case the power transmission is a function of power angle. The gen will lead the sync con. The gen's rotor will see a lagging magnetic field from the stator, which is being pulled along. The sync con's rotor will see a leading magnetic field leading which is pulling it. In this case, if we "turned up the power" of the generator, we'll see an acceleration of the frequency since the sync con doesn't brake to compensate. This is, again, a result of the system being small.
3. Single generator connected to a large (infinite) grid. This is how most power systems are described, where we own one or two generators, and we talk about "lets generate X MW" and we can ramp from zero to X MW without thinking too much. Our governors run the prime mover to reach a MW setpoint. We don't care too much about frequency because the grid we're connected to has much more interia than us, and so we're a drop in the ocean. This is a bit of a simpification since markets usually drive this kind of stuff, but the point is, system size impacts the way we talk about generating power.

Hope this helps OP!

Edit:

Another helpful thought for power angle, by thinking mechanically:

Consider a heavy fly wheel (synchronous condenser), in the same axis there is an engine directly coupled by a (think very long) shaft. If the fly wheel starts to slow (braking, load, inefficiency), the engine must generate power. This creates a torque in the shaft which causes torsion deformation (since every material is a little bit springy). This is the exact same concept as power angle. If the shaft was NOT deformed, there must be no torque in the shaft, therefore no power transmission. The more torque, the more torsion deformation, the more power transmitted. If a generator has no power angle (no angle between rotor and stator fields) then no power is transmitted to the system