Practical engineering has a great video about power black starts that give some insight into this complicated machine.
PS One of the answers in the SO thread mentions JET in the UK. I spent a few summers there as an electrical engineering student (it's home to the MAST and JET fusion reactors). When the JET tokamak ignites a plasma, it can't sustain it for very long (we are not yet at the point of extracting enough energy to sustain the reaction). As a result they need to ignite the plasma and keep it hot. They can't do it for more than 1-10 seconds. During that time, they draw massive amounts of power - they're permitted to draw up to 1% of the UK's capacity for a short period, whilst they simultaneously dump all the energy stored in two gigantic flywheel generators housed in a nearby building. I've never been there when the flywheels are running but I've climbed around beneath them. There's nothing quite like massive engineering :)
Other problem is that for renewables to be profitable you want to push all the energy out all the time, especially in peak. Even now solar installation users have problem with that when there is too many small solar installations installed on same street the voltage goes too high and the inverters just trip and stop pushing the power to the grid, losing owner money.
We just need to have more cheaper storage solutions. Technically utilities could just put a bunch of batteries near concentration of residential solar and just basically sell the service of "storing the kilowatts" to them (say "you can receive 80% of what you put into it in next 48 hours"), all while having the capacity to use that stored joules in case a peak needs to be handled
Generator spins at 50hz Motor spins at 50hz
Add load to the motor, it starts slowing down both itself and the generator. Generator governor increases input energy, frequency goes up. Both are in phase all the time.
Remove load, both start spinning faster. Reduce governor to regulate it to 50hz again.
It's a bit harder with inverters but the idea is similar, you follow the grid phase and if you want to send energy to the grid you will be slightly early to the grid phase and if you want to take energy from the grid your phase will slightly lag.
The difficulty is not "keeping them in phase", that just happens (aside from initial connection), it's the whole load prediction and handling, when to tell which plant to start producing more or less energy, with variety of plants having shorter or longer ramp-up/down periods
The most difficult bit for an operator is to make sure the generator is synchronized before they actually connect it. Only after it's synchronized can they start actually feeding power in.
You can't just turn these things on and off at will like a regular motor. To extend the gears analogy, gears need to be synchronised before they can engage - just like synchronous AC generators.
[0] https://arstechnica.com/tech-policy/2021/05/texas-power-outa...